If the ratio of the component of weight along the road to the component of weight into the road is very large, it means that the horizontal component of the weight of the car is too large
Let's solve the problem step by step:1. A car makes a turn on a banked road. If the road is banked at 10°, show that a vector parallel to the road is (cos 10°, sin 10°).
Since the road is banked, it means the road is inclined with respect to the horizontal. Therefore, the horizontal component of the weight of the car provides the centripetal force that keeps the car moving along the curved path.The horizontal component of the weight of the car is equal to the weight of the car times the sine of the angle of inclination.
Therefore, if the weight of the car is 2000 kg, then the horizontal component of the weight of the car is: Horizontal component of weight = 2000 × sin 10°= 348.16 N (approx)2. If the car has weight 2000 kilograms, find the component of the weight vector along the road vector. This component of weight provides a force that helps the car turn.
The component of the weight vector along the road vector is given by: Weight along the road = 2000 × cos 10°= 1963.85 N (approx)
The ratio of the component of weight along the road to the component of weight into the road is given by: Weight along the road / weight into the road= (2000 × cos 10°) / (2000 × sin 10°)= cos 10° / sin 10°= 0.1763 (approx)
Therefore, the ratio of the component of weight along the road to the component of weight into the road is approximately 0.1763.3.
If the ratio of the component of weight along the road to the component of weight into the road is very small, it means that the horizontal component of the weight of the car is not large enough to provide the necessary centripetal force to keep the car moving along the curved path. Therefore, the car may slide or skid off the road.
This is dangerous. If the ratio of the component of weight along the road to the component of weight into the road is very large, it means that the horizontal component of the weight of the car is too large. Therefore, the car may experience excessive frictional forces, which may cause the tires to wear out quickly or even overheat. This is also dangerous.
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Assuming a 10% reserve requirement. If Bank A sells a $100 security to the Fed, which one of the statements is true? Why the statement is true? (Note: only one statement is true, the wrong statements are not required to be explained.) A) Aggregate reserve in the banking system increase by $100, but the change of monetary supply is uncertain. B) Monetary base has no change if all banks hold no excess reserve. C) Money supply will decrease by $1000 according to multiplication effect. D) Monetary base decreases while money supply increases if Bank A does not lend. beta 段落格式• 字体 字号 B I U A. = ΞΞΩππ
The statement that is true in this scenario is D) Monetary base decreases while money supply increases if Bank A does not lend. In this case, when Bank A sells a $100 security to the Fed, the monetary base decreases because the Fed pays Bank A for the security with reserves.
This reduces the amount of reserves held by Bank A, leading to a decrease in the monetary base. However, if Bank A does not lend out the reserves it receives from the Fed, the money supply remains unchanged.
The reason the statement is true is because the monetary base consists of the currency in circulation and the reserves held by banks at the central bank. When Bank A sells the security to the Fed, it reduces its reserves, which are part of the monetary base. However, if Bank A holds onto the reserves instead of lending them out, the money supply does not increase. Money supply depends on the lending and borrowing activities of banks, and in this scenario, Bank A's decision not to lend prevents an increase in the money supply.
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Consider the following bivariate regression model: Y₁ =B -B (2) + +24, for a given random sample of observations ((Y, X). The regressor is stochastic, whose sample variance is not 0, and X, 0 for all i. We may assume E(X) = 0, where X = (X1,..., Xn). (a) (5 marks) is the following estimator B = -1 X₁Y₂ ΣΥ2 an unbiased estimator for B? Hint: in your answer you need to treat , as a random variable, carefully derive E[BX] first! (b) (3 marks) You are advised that an unbiased estimator for B is given by Σ(*) B = Discuss how you can obtain this estimator. Is this estimator BLUE? Provide suitable arguments to support your answers.
(a) The estimator B = -1/X₁Σ(Y₂²) is unbiased for B.
(b) To evaluate if the estimator Σ(*) B is BLUE, more information is needed about other unbiased estimators and their variances.
(a) To determine if the estimator B = -1/X₁Σ(Y₂²) is unbiased for B, we need to calculate E[B|X].
First, let's derive the expression for E[B|X]:
E[B|X] = E[-1/X₁Σ(Y₂²)]
= -1/X₁ΣE(Y₂²)
Since Y₂ is the dependent variable in the regression model, we can express it as:
Y₂ = B₀ + B₁X + ε
Taking the expectation of Y₂²:
E(Y₂²) = E[(B₀ + B₁X + ε)²]
= E[B₀² + 2B₀B₁X + B₁²X² + 2B₀ε + 2B₁Xε + ε²]
= B₀² + 2B₀B₁E(X) + B₁²E(X²) + 2B₀E(ε) + 2B₁XE(ε) + E(ε²)
= B₀² + B₁²E(X²) + E(ε²)
Since E(X) = 0 and E(ε) = 0, the expression simplifies to:
E(Y₂²) = B₀² + B₁²E(X²) + E(ε²)
Substituting this back into the expression for E[B|X]:
E[B|X] = -1/X₁Σ(B₀² + B₁²E(X²) + E(ε²))
= -1/X₁Σ(B₀² + B₁²E(X²) + Var(ε))
= -1/X₁Σ(B₀² + B₁²E(X²) + σ²) (since Var(ε) = σ²)
Now, we can determine if E[B|X] equals B to determine if the estimator B = -1/X₁Σ(Y₂²) is unbiased for B. If E[B|X] = B, then the estimator is unbiased.
(b) The proposed unbiased estimator Σ(*) B can be obtained by summing the individual estimates for B from each observation in the sample.
To determine if this estimator is BLUE (Best Linear Unbiased Estimator), we need to check if it satisfies the properties of linearity, unbiasedness, and minimum variance among all unbiased estimators.
- Linearity: The estimator Σ(*) B is linear since it is obtained by summing the individual estimates.
- Unbiasedness: The estimator is unbiased if the expected value of the estimator equals the true parameter value. We need to calculate E[Σ(*) B] and check if it equals B.
- Minimum variance: To establish minimum variance, we need to compare the variance of the estimator Σ(*) B with the variances of other unbiased estimators and determine if it has the smallest variance among them.
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Find the general solution of the differential equation. Just choose any 2
a. yy' = -8 cos (πx)
b. √(1-4x^(2)y') = x
c. y ln x -xy' = 0
The general solutions to the given differential equations are as follows:
a. For the differential equation yy' = -8 cos(πx), the general solution can be found by separating variables and integrating. After integrating, we obtain the solution y^2/2 = -8 sin(πx)/π + C, where C is the constant of integration.
b. For the differential equation √(1-4x^2)y' = x, we can solve by separating variables and integrating. By integrating, we find √(1-4x^2) = (x^2/2) + C, where C is the constant of integration.
a. To solve the first differential equation yy' = -8 cos(πx), we can separate variables by writing it as ydy = -8 cos(πx)dx. Integrating both sides gives y^2/2 = -8 sin(πx)/π + C, where C is the constant of integration. To find the general solution, we can multiply both sides by 2 and take the square root, yielding y = ±√(-16 sin(πx)/π + 2C).
b. For the differential equation √(1-4x^2)y' = x, we can start by separating variables to obtain √(1-4x^2)dy = xdx. Integrating both sides gives the equation √(1-4x^2) = (x^2/2) + C, where C is the constant of integration. To simplify the equation, we square both sides, which leads to 1-4x^2 = (x^2/2 + C)^2. Solving for y, we get y = ±√[(x^2/2 + C)^2 - 1 + 4x^2]. This represents the general solution to the differential equation.
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A random sample of 40 adults with no children under the age of 18 years results in a mean daily leisure time of 5.65 hours, with a standard deviation of 2.43 hours. A random sample of 40 adults with children under the age of 18 results in a mean daily leisure time of 4.37 hours, with a standard deviation of 1.73 hours. Construct and interpret a 90% confidence interval for the mean difference in leisure time between adults with no children and adults with children. (u1 - u2)
The interpretation is that we are 90% confident that the true difference in the mean daily leisure time between adults with no children and adults with children lies between 0.451 and 2.109 hours.
The given data are: For adults with no children under the age of 18 years, Mean = 5.65 hours Standard deviation = 2.43 hours Sample size, n1 = 40 For adults with children under the age of 18, Mean = 4.37 hours Standard deviation = 1.73 hours Sample size, n2 = 40 The formula to calculate the 90% confidence interval for the difference between two means can be given as:\[\left( {{\bar x}_1} - {{\bar x}_2} \right) \pm {t_{\frac{\alpha }{2},n_1 + {n_2} - 2}}\sqrt {\frac{{s_1^2}}{n_1} + \frac{{s_2^2}}{n_2}}\]where,${{\bar x}_1}$ is the sample mean for group 1,${{\bar x}_2}$ is the sample mean for group 2,${{s_1}}$ is the sample standard deviation for group 1,${{s_2}}$ is the sample standard deviation for group 2,$\alpha$ is the level of significance,$n_1$ is the sample size for group 1,$n_2$ is the sample size for group 2,and $t_{\frac{\alpha }{2},n_1 + {n_2} - 2}$ is the t-value from the t-distribution with (n1 + n2 – 2) degrees of freedom.
Let's calculate the confidence interval as follows: Mean difference, $\left( {{\bar x}_1} - {{\bar x}_2} \right)$= 5.65 − 4.37= 1.28 hours Sample standard deviation for group 1, ${s_1}$ = 2.43 hours Sample standard deviation for group 2, ${s_2}$ = 1.73 hours Sample size for group 1, ${n_1}$ = 40Sample size for group 2, ${n_2}$ = 40 Degree of freedom = $n_1 + n_2 - 2$= 40 + 40 – 2= 78$\alpha$ = 0.1 (90% confidence interval, $\alpha$ = 1 – 0.9 = 0.1)Using the t-table or calculator with the given values, we get:$t_{\frac{\alpha }{2},n_1 + {n_2} - 2}$ = t0.05, 78 = 1.665 (approximately)Substituting the given values in the formula, we get:\[\left( {{\bar x}_1} - {{\bar x}_2} \right) \pm {t_{\frac{\alpha }{2},n_1 + {n_2} - 2}}\sqrt {\frac{{s_1^2}}{n_1} + \frac{{s_2^2}}{n_2}}\] = $1.28 \pm 1.665\sqrt {\frac{{2.43^2}}{40} + \frac{{1.73^2}}{40}}$= $1.28 \pm 0.829$= (0.451, 2.109)
Therefore, the 90% confidence interval for the mean difference in leisure time between adults with no children and adults with children is (0.451, 2.109) hours.
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We can also say that adults with no children have, on average, between 0.50 hours and 2.06 hours more leisure time per day than adults with children under the age of 18.
The 90% confidence interval for the mean difference in leisure time between adults with no children and adults with children is (-1.23, -0.04).
We are to construct a 90% confidence interval for the mean difference in leisure time between adults with no children and adults with children.
We are given the following information:
u1 = mean daily leisure time of adults with no children
= 5.65 hours
σ1 = standard deviation of daily leisure time of adults with no children
= 2.43 hours
n1 = sample size of adults with no children
= 40
u2 = mean daily leisure time of adults with children
= 4.37 hours
σ2 = standard deviation of daily leisure time of adults with children
= 1.73 hours
n2 = sample size of adults with children
= 40
We can find the standard error (SE) of the difference in means as follows:
SE = sqrt [ (σ1^2 / n1) + (σ2^2 / n2) ]
SE = sqrt [ (2.43^2 / 40) + (1.73^2 / 40) ]
SE = sqrt (0.1482 + 0.0752)
SE = sqrt (0.2234)
SE = 0.4726
We can now use the formula for a confidence interval of the difference in means as follows:
CI = ( (u1 - u2) - E , (u1 - u2) + E )
where
E = z*SE and z* is the z-score for the level of confidence.
Since we want a 90% confidence interval, we look for the z-score that corresponds to the middle 90% of the normal distribution, which is found using a z-table or calculator.
For a 90% confidence level, the z* value is 1.645,
so:E = 1.645 * 0.4726E = 0.7779
Plugging in the values, we have:CI = ( (5.65 - 4.37) - 0.7779 , (5.65 - 4.37) + 0.7779 )CI = ( 1.28 - 0.78, 1.28 + 0.78 )CI = ( 0.50, 2.06 )
The 90% confidence interval for the mean difference in leisure time between adults with no children and adults with children is (0.50, 2.06).
This means that we are 90% confident that the true mean difference in leisure time between the two groups of adults falls between 0.50 hours and 2.06 hours.
Since the interval does not include zero, we can conclude that the difference in means is statistically significant at the 0.10 level. We can also say that adults with no children have, on average, between 0.50 hours and 2.06 hours more leisure time per day than adults with children under the age of 18.
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point) Suppose that the matrix A has repeated eigenvalue with the following eigenvector and generalized eigenvector: A = 3 with eigenvector ? and generalized eigenvector w Write the solution to the linear system = ' = Av in the following forms. A. In eigenvalueleigenvector form: %0) _ [3l In fundamental matrix form: x(t) y(t) 4e^ (3t) 3+4tje"(3t) 2e^(3t) C.As two equations: (write "c1 and c2" for C1 and C2 x(t) 4e^(3t)(c1+c2(3/4+t)) Note: if you are feeling adventurous You could use other eigenvectors like 4 € and other generalized eigenvectors like w 3v_ Just remember that if you change U, You must also change W for its fundamenta solution!
To solve the linear system given by x' = Ax, where A is a matrix with a repeated eigenvalue, we can express the solution in different forms.
A. In eigenvalue-eigenvector form:
The eigenvalue is 3, and the eigenvector associated with it is represented as v. So, the solution can be written as x(t) = e^(3t)v.
B. In fundamental matrix form:
The fundamental matrix is constructed using the eigenvectors and generalized eigenvectors. In this case, the fundamental matrix is:
[x(t)] [4e^(3t) 3+4t] [c1]
[y(t)] = [2e^(3t)] * [ 1 ] * [c2]
C. As two equations:
Another way to represent the solution is by writing it as two separate equations:
x(t) = 4e^(3t)(c1 + c2(3/4 + t))
y(t) = 2e^(3t)(c1 + c2(1))
Here, c1 and c2 are constants that depend on the initial conditions of the system.
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Find the sum of the first four terms of a geometric sequence with a₁ = -1 and r = 3
The sum of the first four terms of the given geometric sequence is 40.
To find the sum of the first four terms of a geometric sequence with a first term (a₁) of -1 and a common ratio (r) of 3, we can use the formula for the sum of a finite geometric series:
S₄ = a₁ * (1 - r⁴) / (1 - r),
where S₄ represents the sum of the first four terms.
Substituting the given values into the formula, we have:
S₄ = -1 * (1 - 3⁴) / (1 - 3).
Calculating the numerator and denominator separately:
Numerator:
1 - 3⁴ = 1 - 81 = -80.
Denominator:
1 - 3 = -2.
Now, substituting the numerator and denominator back into the formula:
S₄ = -1 * (-80) / (-2) = 80 / 2 = 40.
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While grading her students' most recent quiz on equation solving, Mrs. Jones calculated that approximately forty percent of her students answered question number 14 with multiple choice option B, while the other sixty percent answered A or C.Question #14 from Mrs. Jones's students' most recent quiz:14) Solve the single variable equation for n .3(-n+4) + 5n =2na.n = 3b.no solutionc.infinitely many solutionsPart 1: Use inverse operations and rules of equation solving to determine the correct answer to Mrs. Jones's quiz question number 14. Include all of your work in your final answer.Part 2: Use complete sentences to compare the similarities and differences of each of the multiple choice answer options A-C. In your answer, rationalize why a student would choose each of the options as the correct answer.
C) There are infinite solutions for the given equation
Part 1:
For solving the single variable equation of #14 in the quiz,
3(-n+4) + 5n = 2n
Distribute the 3 on the left side of the equation.
-3n + 12 + 5n = 2n
Simplify.
2n + 12 = 2n
Combine like terms.
-2n -2n2n + 12 - 2n = 02n + 12 = 0
Now, subtract 12 on both sides of the equation.
2n + 12 - 12 = -12n = -6
Therefore, option B has the correct answer.
Part 2:Options A-C are as follows:
A. n = 3B. No solution
C. Infinitely many solutions
A. n = 3:
This option indicates that n is equal to 3, which is incorrect. A student can choose this option as a correct answer by adding 3 to both sides of the equation, instead of subtracting 2n from both sides of the equation.
B. No solution:
This option indicates that the solution for the given equation is not possible. A student can choose this option as a correct answer by making a calculation error while solving the equation or misunderstanding the concept of the given equation.
C. Infinitely many solutions:
This option indicates that there are infinite solutions for the given equation. A student can choose this option as a correct answer by adding 2n on both sides of the equation and then simplifying the equation. However, the equation is an identity, which means that the solution is not equal to n but to all real numbers.
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Suppose that the distance of fly balls hit to the outfield (in baseball) is normally distributed with a mean of 264 feet and a standard deviation of 43 feet. Let X be the distance in feet for a fly ball.
Given that the distance of fly balls hit to the outfield follows a normal distribution with a mean of 264 feet (μ = 264) and a standard deviation of 43 feet (σ = 43).
we can denote X as the random variable representing the distance in feet for a fly ball.
Therefore, we can express this mathematically as:
X ~ N(264, 43^2)
This notation indicates that X is normally distributed with a mean of 264 and a variance of 43^2.
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15 - 1/2x = 7 solve the equation
Suppose an ANOVA experiment is comparing means across 4 different categories. If one were to perform the necessary paired t tests for all 4 categories with an alpha = .01, what would the overall Type I error probability be?
The overall Type I error probability would be approximately 0.0396 or 3.96%.
To determine the overall Type I error probability when performing multiple paired t-tests across 4 different categories, we need to consider the concept of familywise error rate (FWER). The FWER is the probability of making at least one Type I error among all the tests conducted.
In this case, we are conducting 4 paired t-tests, each with a significance level of α = 0.01. The Type I error rate for each individual test is α = 0.01.
To calculate the overall Type I error probability, we use the formula for FWER:
Overall Type I error probability = 1 - (1 - α)^k
Where k is the number of tests conducted.
In this scenario, k = 4 (since we are performing 4 paired t-tests). Substituting the values into the formula, we have:
Overall Type I error probability = 1 - (1 - 0.01)^4 = 1 - 0.99^4 ≈ 0.0396
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A national survey indicated that 30% of adults conduct their banking online. It also found that 60% are under the age of 50, and that 15% are under the age of 50 and conduct their banking online.
a) What percentage of adults do not conduct their banking online?
b) What type of probability is the 15% mentioned above?
c) Construct a contingency table showing all joint and marginal probabilities.
d) What is the probability that an individual conducts banking online given that the individual is under the age of 50?
e) Are Banking online and Age independent?
a) The adults do not do banking online = 70%. b) 15% mentioned above is joint probability. c) marginal probabilities are calculated. d) The probability that an individual conducts banking online under the age of 50 is 25%. e) No, Banking online and Age are not independent variables.
a) Let's consider that we have 100 adults in total.
According to the survey, 30 % of them do banking online.
Then, the rest of the adults do not do banking online:100 - 30 = 70, which is 70 % of the adults.
b) 15% mentioned above is joint probability.
Joint probability is the probability of two or more events happening together. The event of conducting banking online and being under the age of 50 are happening together.
c) Construct a contingency table showing all joint and marginal probabilities.
The row represents age, and the column represents the choice of banking. Joint probabilities are filled in the cells, and marginal probabilities are calculated.
d) The probability that an individual conducts banking online given that the individual is under the age of 50 is 0.25 or 25%.
We need to find P(Online Banking | Under 50)
P(Online Banking | Under 50) = P(Online Banking and Under 50) / P(Under 50)P(Online Banking | Under 50) = 0.15 / 0.6 = 0.25 or 25%
e) No, Banking online and Age are not independent variables
If two events are independent, then the occurrence of one event does not affect the probability of the occurrence of the other event.
But, if two events are dependent, then the occurrence of one event affects the probability of the occurrence of the other event.
Here, Age and Online Banking are dependent variables. The occurrence of one event (Age) affects the probability of the occurrence of the other event (Online Banking).
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Let Y₁, Y, Y, denote a random sample from pdf: fo) = ((0+1)y 0
The CDF for the random variable Y is F(y) = y² + y for 0 ≤ y ≤ 1.
The probability that Y is greater than 0.5 is 0.25.
The expected value (mean) of Y is 7/6 and the variance of Y is -1/36.
We have,
Calculate the cumulative distribution function (CDF) for the random variable Y.
The cumulative distribution function (CDF), denoted as F(y), represents the probability that the random variable Y takes on a value less than or equal to y.
For the given pdf, we can calculate the CDF as follows:
F(y) = ∫[0, y] (2t + 1) dt
To find the integral, we integrate the expression (2t + 1) with respect to t from 0 to y.
Simplifying the integral gives us:
F(y) = [t² + t] evaluated from 0 to y
= (y² + y) - (0² + 0)
= y² + y
Therefore, the CDF for the random variable Y is F(y) = y² + y for 0 ≤ y ≤ 1.
Determine the probability that Y is greater than 0.5.
To find the probability that Y is greater than 0.5, we can use the CDF:
P(Y > 0.5) = 1 - P(Y ≤ 0.5)
= 1 - F(0.5)
= 1 - (0.5² + 0.5)
= 1 - 0.25 - 0.5
= 0.25
Therefore, the probability that Y is greater than 0.5 is 0.25.
Find the expected value (mean) and variance of the random variable Y.
The expected value (mean) of a random variable Y can be calculated using the formula:
E(Y) = ∫[0, 1] y x (2y + 1) dy
Integrating the expression y x (2y + 1) gives us:
E(Y) = ∫[0, 1] (2y² + y) dy
= [2/3 x y³ + 1/2 x y²] evaluated from 0 to 1
= (2/3 x 1³ + 1/2 x 1²) - (2/3 x 0³ + 1/2 x 0²)
= 2/3 + 1/2
= 7/6
Therefore, the expected value (mean) of the random variable Y is E(Y) = 7/6.
To calculate the variance of the random variable Y, we can use the formula:
Var(Y) = E(Y²) - [E(Y)]²
The term E(Y²) can be found by evaluating the integral:
E(Y^2) = ∫[0, 1] y² x (2y + 1) dy
Integrating y² x (2y + 1) gives us:
E(Y²) = ∫[0, 1] (2y³ + y²) dy
[tex]= 1/2 \times y^4 + 1/3 \times y^3[/tex]
evaluated from 0 to 1
[tex]= (1/2 * 1^4 + 1/3 * 1^3) - (1/2 * 0^4 + 1/3 * 0^3)[/tex]
= 1/2 + 1/3
= 5/6
Substituting these values into the variance formula:
[tex]Var(Y) = E(Y^2) - [E(Y)]^2[/tex]
= 5/6 - (7/6)²
= 5/6 - 49/36
= -1/36
Therefore, the variance of the random variable Y is Var(Y) = -1/36.
Thus,
The CDF for the random variable Y is F(y) = y² + y for 0 ≤ y ≤ 1.
The probability that Y is greater than 0.5 is 0.25.
The expected value (mean) of Y is 7/6 and the variance of Y is -1/36.
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The complete question:
Let Y₁, Y₂, Y₃ be a random sample from a probability density function (pdf) given by:
Find the probability that Y is greater than 0.5.
Find the CDF of the random variable.
f(y) = (2y + 1), for 0 ≤ y ≤ 1
Find the expected value and variance of Y.
It is known that 20% of the people in a certain human population are female. The experiment is to select a committee consisting of two individuals at random. Let X be a random variable giving the number of females in the committee. Find the variance. A box contains 100 cards; 40 of which are labeled with the number 5 and the other cards are labeled with the number 10. Two cards were selected randomly with replacement and the number appeared on each card was observed. Let X be a random variable giving the total sum of the two numbers. Find P(X> 10)
To find the variance of the random variable X, which represents the number of females in a committee of two individuals randomly selected from a human population where 20% are female, we can use the binomial distribution.
The variance of a binomial distribution is given by the formula Var(X) = np(1 - p), where n is the number of trials and p is the probability of success.
In this case, n = 2 (as we are selecting a committee of two individuals) and p = 0.2 (as the probability of selecting a female is 20%).
Therefore, the variance of X is Var(X) = 2 * 0.2 * (1 - 0.2) = 0.32.
Hence, the variance of the random variable X is 0.32.
For the second part of the question, let's consider the random variable X, which represents the total sum of the numbers observed on two cards randomly selected with replacement from a box containing 100 cards (40 labeled with 5 and the rest labeled with 10).
To find P(X > 10), we need to calculate the probability of getting a sum greater than 10.
Let's consider the possible outcomes when selecting two cards:
Both cards are labeled 5: The sum is 5 + 5 = 10.
One card is labeled 5 and the other is labeled 10: The sum is 5 + 10 = 15.
Both cards are labeled 10: The sum is 10 + 10 = 20.
We are interested in the probability of getting a sum greater than 10, which is P(X > 10). In this case, only one outcome satisfies this condition, which is when the sum is 15.
Since the cards are selected with replacement, each selection is independent, and the probabilities can be multiplied together. The probability of selecting a card labeled 5 is 40/100 = 0.4, and the probability of selecting a card labeled 10 is 60/100 = 0.6.
Therefore, P(X > 10) = P(X = 15) = P(5 and 10) + P(10 and 5) = (0.4 * 0.6) + (0.6 * 0.4) = 0.24 + 0.24 = 0.48.
Hence, the probability that the sum of the numbers observed on the two cards is greater than 10 is 0.48.
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Each sweat shop worker at a computer factory can put together 4.4 computers per hour on average with a standard deviation of 0.8 computers. 49 workers are randomly selected to work the next shift at the factory. Round all answers to 4 decimal places where possible and assume a normal distribution. a. What is the distribution of X? X-N b. What is the distribution of a?a-N c. What is the distribution of Σa? Σa-N d. If one randomly selected worker is observed, find the probability that this worker will put together between 4.3 and 4.4 computers per hour. e. For the 49 workers, find the probability that their average number of computers put together per hour is between 4.3 and 4.4. f. Find the probability that a 49 person shift will put together between 210.7 and 215.6 computers per hour. g. For part e) and f), is the assumption of normal necessary? No Yes h. A sticker that says "Great Dedication will be given to the groups of 49 workers who have the top 20% productivity. What is the least total number of computers produced by a group that receives a sticker? computers per hour (round to the nearest computer)
a. The distribution of X is normal (X ~ N) as we assume workers' productivity follows a normal distribution.
b. The distribution of a is normal (a ~ N) as we assume the average number of computers per worker is normally distributed.
c. The distribution of Σa is normal (Σa ~ N) because the sum of normally distributed variables is also normally distributed.
d. The probability that a randomly selected worker puts together between 4.3 and 4.4 computers per hour can be calculated using the normal distribution.
e. The probability that the average number of computers per hour for the 49 workers is between 4.3 and 4.4 can be calculated using the distribution of sample means.
f. The probability that a 49-person shift puts together between 210.7 and 215.6 computers per hour can be calculated using the distribution of the sum of their productivity.
g. The assumption of a normal distribution is necessary for parts e) and f) due to the central limit theorem.
h. The least total number of computers produced by a group receiving a sticker can be found by calculating the cutoff value corresponding to the top 20% of productivity using z-scores and the normal distribution.
a. The distribution of X is normal (X ~ N) because we are assuming that the workers' productivity follows a normal distribution. This means that the number of computers each worker can put together per hour is normally distributed.
b. The distribution of a is also normal (a ~ N) because we are assuming that the workers' productivity, represented by variable 'a', follows a normal distribution. This is based on the assumption that the average number of computers each worker puts together per hour is normally distributed.
c. The distribution of Σa is also normal (Σa ~ N) because the sum of normally distributed variables is also normally distributed. In this case, Σa represents the total productivity of all 49 workers in a shift, and it follows a normal distribution.
d. To find the probability that a randomly selected worker will put together between 4.3 and 4.4 computers per hour, we can calculate the area under the normal distribution curve between those two values. This can be done using statistical software or standard normal distribution tables.
e. To find the probability that the average number of computers put together per hour for the 49 workers is between 4.3 and 4.4, we can calculate the area under the normal distribution curve for the distribution of sample means. The standard deviation of the sample means can be obtained by dividing the standard deviation of individual workers (0.8 computers) by the square root of the sample size (√49 = 7). We can then use this standard deviation and the sample mean to calculate the probability.
f. To find the probability that a 49-person shift will put together between 210.7 and 215.6 computers per hour, we need to consider the distribution of the sum of the 49 workers' productivity. The mean of the sum is the product of the average productivity per worker (4.4 computers) and the number of workers (49). The standard deviation of the sum can be obtained by multiplying the standard deviation of individual workers (0.8 computers) by the square root of the sample size (√49 = 7). We can then use this mean and standard deviation to calculate the probability.
g. The assumption of a normal distribution is necessary for both parts e) and f) because we are dealing with averages and sums of random variables. The central limit theorem states that the distribution of sample means and sums tends to be approximately normal, regardless of the shape of the population distribution, as long as the sample size is sufficiently large.
h. To determine the least total number of computers produced by a group that receives a sticker, we need to find the cutoff value corresponding to the top 20% of productivity. This can be done by finding the z-score associated with the 80th percentile of the normal distribution. We can then multiply this z-score by the standard deviation of individual workers (0.8 computers) and add it to the mean productivity per worker (4.4 computers) to obtain the least total number of computers required.
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Evaluate the sum, difference, product, or quotient of two functions. Let,f(x) = x² - 4x and g(x) = x + 13 find the following: a. (f+g)(-2) b. (f-g)(-2) c. f(x)-g(x)
The problem requires us to evaluate the sum, difference, and product of two functions, f(x) = x² - 4x and g(x) = x + 13, as well as the difference between the two functions. We need to find the values of these expressions at a specific input, which is -2 in this case.
a. To find (f+g)(-2), we substitute -2 into both functions and add the results.
(f+g)(-2) = f(-2) + g(-2) = ((-2)² - 4(-2)) + (-2 + 13) = (4 + 8) + 11 = 12 + 11 = 23.
b. To find (f-g)(-2), we substitute -2 into both functions and subtract the results.
(f-g)(-2) = f(-2) - g(-2) = ((-2)² - 4(-2)) - (-2 + 13) = (4 + 8) - 11 = 12 - 11 = 1.
c. To find f(x) - g(x), we subtract the second function from the first function without substituting a specific value.
f(x) - g(x) = (x² - 4x) - (x + 13) = x² - 4x - x - 13 = x² - 5x - 13.
Therefore, the results are:
a. (f+g)(-2) = 23
b. (f-g)(-2) = 1
c. f(x) - g(x) = x² - 5x - 13.
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Use the series to evaluate the limit
lim y -> 0 (arctan(y) - sin y)/(y ^ 3 * cos y)
as y approaches 0, the limit is 0.
To evaluate the limit as y approaches 0 of (arctan(y) - sin(y))/(y^3 * cos(y)), we can use the Taylor series expansions of the functions involved. Thus, we can evaluate the limit by identifying the coefficient of the highest power of y in the resulting series.
To evaluate the limit as y approaches 0 of (arctan(y) - sin(y))/(y^3 * cos(y)), we can expand the functions arctan(y), sin(y), and cos(y) using their Taylor series expansions. The Taylor series expansion for arctan(y) is y - (y^3)/3 + (y^5)/5 - ..., the expansion for sin(y) is y - (y^3)/6 + (y^5)/120 - ..., and the expansion for cos(y) is 1 - (y^2)/2 + (y^4)/24 - .... By substituting these series expansions into the given expression, we get:
[(y - (y^3)/3 + (y^5)/5 - ...) - (y - (y^3)/6 + (y^5)/120 - ...)] / [(y^3 * (1 - (y^2)/2 + (y^4)/24 - ...))]
Simplifying the expression, we get:
[(2y^3)/3 + O(y^5)] / [(y^3 - (y^5)/2 + O(y^7))]
Here, O(y^n) represents terms of higher order than y^n. We can neglect these terms as y approaches 0. The leading term in the simplified expression is (2y^3)/3. Therefore, as y approaches 0, the limit is 0.
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6) Write the face value and place value of each digit of decimal number (10 Marks)
a) 2.109
b) 50004
7) What is the largest seven digit decimal number? (5 Marks)
8) What it the smallest seven-digit decimal number? (5 Marks)
9) What is the largest five-digit binary number? (5 Marks)
10) List the first 12 powers of 2 and the first 4 powers of 8. Can you find a relationship between the two systems? It is this relationship that makes octal numbers important for computers (5 Marks)
11) Which is larger? (5 Marks)
C798 16 or B6021 16
6) Decimal Number Digit Analysis:
a) For the decimal number 2.109:
- The face value and place value of the digit 2 are both 2.
- The face value and place value of the digit 1 before the decimal point are both 1.
- The face value and place value of the digit 0 after the decimal point are both 0.
- The face value and place value of the digit 9 after the decimal point are both 9.
b) For the decimal number 50004:
- The face value and place value of the digit 5 are both 5.
- The face value and place value of the digit 0 in the tens place are both 0.
- The face value and place value of the digit 0 in the ones place are both 0.
- The face value and place value of the digit 0 in the thousands place are both 0.
- The face value and place value of the digit 0 in the ten thousands place are both 0.
- The face value and place value of the digit 4 in the hundred thousands place are both 4.
7) The largest seven-digit decimal number is 9,999,999.
8) The smallest seven-digit decimal number is 1,000,000.
9) The largest five-digit binary number is 31 in decimal representation. In binary, it is represented as 11111.
10) Powers of 2 and Powers of 8:
- Powers of 2: 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096.
- Powers of 8: 1, 8, 64, 512.
The relationship between the two systems is that each digit in an octal number represents three bits in binary. Octal numbers are base-8, and binary numbers are base-2. Since computers use binary as their fundamental number system, octal numbers are important for representing and working with groups of bits efficiently.
11) Comparing Hexadecimal Numbers:
- C798₁₆ (base 16) is larger than B602₁₆ (base 16).
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solve the triangle(s) with the understanding that a is opposite
angle
A, side b is opposite angle B and c is opposite angle, C round
angles to the nearest tenth of
a degree and sides to three signific
The triangle is solved and the sides and angles are rounded to three significant figures as follows.a = 9.9 cmb = 26.0 cmc = 20.9 cmA = 20.8°B = 65.5°C = 94.2°. A triangle can be solved by using the law of sines and the law of cosines to solve for missing sides and angles.
In a triangle ABC, let angle A be opposite side a, angle B be opposite side b, and angle C be opposite side c. Also, let the values of sides a, b, and c be known. The triangle can be solved for missing sides and angles by using the following equations.1. Law of Sines:a/sinA = b/sinB = c/sinC2. Law of Cosines:a² = b² + c² - 2bc cos A ; solve for a (two possible solutions) b² = a² + c² - 2ac cos B ; solve for b (two possible solutions) c² = a² + b² - 2ab cos C ; solve for c (two possible solutions)
Given the triangle with a, b, and c opposite angles A, B, and C respectively.Since we know that the sum of all angles in a triangle is 180°, we can find the measure of angle A using the following formula.A + B + C = 180°Substituting the known values, we haveA + 82.7° + 76.5° = 180°A = 180° - 82.7° - 76.5°A = 20.8°Therefore, the measure of angle A is 20.8°.To
⇒ b/sin(65.5°) = 20.9/sin(38°)b = (sin 65.5°/sin 38°) * 20.9b = 26.0 cm
Therefore, the length of side b is 26.0 cm.To find the length of side a, we can use the law of cosines.2bc cos A = b² + c² - a²2(20.9)(12.4) cos 20.8° = 26.0² + 20.9² - a²519.68 cos 20.8° = 1359.21 - a²a² = 1359.21 - 519.68 To check the values, we can verify that the sum of all angles in the triangle is 180°.A + B + C = 20.8° + 65.5° + 94.2° = 180.5°Therefore, the triangle is solved and the sides and angles are rounded to three significant figures as follows.a = 9.9 cmb = 26.0 cmc = 20.9 cmA = 20.8°B = 65.5°C = 94.2°
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Answer the following without graphing: Suppose f(x) = -5(x-4)(x + 7). Which of the following are true? Select all that apply. Aš a-[infinity], f(x) 8x As a →[infinity], f(x) → [infinity] □ Asa-[infinity], f(x) → [infinity] OAST → [infinity], ƒ(1)→ [infinity]
To determine which statements are true regarding the function f(x) = -5(x-4)(x + 7), we need to analyze the behavior of the function as x approaches positive or negative infinity.
Evaluate the function at specific values. a) As x approaches negative infinity (x → -∞), the function f(x) can be analyzed by considering the behavior of the leading term, -5(x-4)(x + 7). As x becomes increasingly negative, both (x-4) and (x + 7) will tend to negative infinity, resulting in a positive value for f(x). Therefore, the statement "As x → -∞, f(x) → 8x" is false. b) As x approaches positive infinity (x → +∞), the function f(x) can be analyzed in a similar manner. Both (x-4) and (x + 7) will tend to positive infinity as x becomes increasingly large, resulting in a negative value for f(x). Hence, the statement "As x → +∞, f(x) → +∞" is false.
c) As x approaches negative infinity (x → -∞), the function f(x) can be analyzed by considering the behavior of the leading term, -5(x-4)(x + 7). Both (x-4) and (x + 7) will tend to negative infinity as x becomes increasingly negative, resulting in a positive value for f(x). Therefore, the statement "As x → -∞, f(x) → -∞" is false. d) The statement "As x → +∞, f(x) → +∞" indicates that as x approaches positive infinity, the function f(x) also tends to positive infinity. However, since the leading term of f(x) is -5(x-4)(x + 7), the function will tend to negative infinity as x approaches positive infinity. Thus, the statement "As x → +∞, f(x) → +∞" is false.
In summary, none of the given statements are true for the function f(x) = -5(x-4)(x + 7). The correct behavior of the function is that as x approaches negative infinity, f(x) tends to a positive value, and as x approaches positive infinity, f(x) tends to negative infinity.
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Select one of the following functions to match the graph.
A. y = (7) x
B. y = (7)-x
C. y = (-7) x
D. y = (-7)-x
Answer:
B
Step-by-step explanation:
The explanation is attached below.
How Did I Do? Let G(x) = 22-25 5-x +x - 3| for x 5. a) As a approaches 5, the quotient gives an indeterminate form of the type
O 1[infinity]
O [infinity]0/0
O 0/0
O [infinity]/[infinity]
O [infinity]-[infinity]
Given that G(x) = 22 - 25(5 - x) + x - 3. As a approaches 5, the quotient gives an indeterminate form of the type 0/0.Hence, the correct answer is O 0/0.
Now, we need to find as x approaches 5, the quotient gives an indeterminate form of the type.
O 1[infinity]O [infinity]0/0O 0/0O [infinity]/[infinity]O
[infinity]-[infinity]
Now, we will solve it:
G(x) = 22 - 25(5 - x) + x - 3
= 22 - 125 + 25x + x - 3
= 5x - 106
Now, using the given function G(x), we need to find the limit of G(x) as x approaches 5.Let x approach 5, thenG(x)
= 5(5) - 106
= 25 - 106
= -81.
Therefore, as a approaches 5, the quotient gives an indeterminate form of the type 0/0.Hence, the correct answer is O 0/0.
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1) Five windows in a house could be either open, ajar, or closed. How many possible states could the windows be?
The number of possible states for the windows in the house can be determined by considering the options for each window.
Since each window can be either open, ajar, or closed, there are three possibilities for each window. Therefore, to find the total number of possible states, we multiply the number of possibilities for each window together.
Since there are five windows in the house, we multiply the number of possibilities (3) by itself five times: 3 × 3 × 3 × 3 × 3 = 243.
Hence, there are 243 possible states for the windows in the house. This means that the windows can be arranged in 243 different combinations of open, ajar, and closed states. Each state represents a unique configuration of the windows, and the total number of states reflects the variety of possible arrangements.
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Consider the initial value problem y' = 2+t-y y (0) = 2. Use the Euler method to approximate y(0.3) by using step size h = 0.1. (Please make sure to write all details of at least 2 steps in your calculation. In particular, the expressions Yn+1 = Yn+h⋅ f(tn, Yn) must be clearly stated with all the numerical values plugged in, for at least the first two steps. The numerical details of the calculation of f(tn, Yn) should also be clearly stated).
Using the Euler method with a step size of h = 0.1, we can approximate the value of y(0.3) for the initial value problem y' = 2+t-y, y(0) = 2.
To approximate the value of y(0.3), we can use the Euler method, which is a simple numerical method for solving ordinary differential equations. In this method, we take small steps (in this case, h = 0.1) and calculate the value of y at each step.
Given the initial condition y(0) = 2 and the differential equation y' = 2+t-y, we can start by evaluating the function f(tn, Yn) at t = 0 and Y = 2. Plugging these values into the equation, we get f(0, 2) = 2 + 0 - 2 = 0.
For the first step, we use the formula Yn+1 = Yn + h * f(tn, Yn). Substituting the known values, we have Y1 = 2 + 0.1 * 0 = 2.
Moving on to the second step, we need to evaluate f(tn, Yn) at t = 0.1 and Y = 2. Plugging these values into the equation, we get f(0.1, 2) = 2 + 0.1 - 2 = 0.1.
Using the Euler method formula again, Y2 = Y1 + h * f(tn, Yn), we have Y2 = 2 + 0.1 * 0.1 = 2.01.
By continuing this process, we can calculate the value of y(0.3) by taking steps of size h = 0.1. However, since we only need to show the details of two steps, the first two approximations Y1 and Y2 are sufficient for this problem.
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Identify the ordered pairs on the unit circle corresponding to each real number r. Write your answer as a simplified fraction, if necessary.
(a) t= - 16π/3 corresponds to the point (x, y) = ___
To find the ordered pair (x, y) on the unit circle corresponding to the real number r = -16π/3, we can use the trigonometric values on the unit circle.
On the unit circle, the angle -16π/3 corresponds to a rotation of 16π/3 in the clockwise direction from the positive x-axis. To determine the corresponding point (x, y), we can refer to the values of sine and cosine for this angle. Recall that on the unit circle, the x-coordinate represents the cosine value and the y-coordinate represents the sine value. Using the angle -16π/3, we can find the corresponding point on the unit circle.
The cosine of -16π/3 is given by cos(-16π/3) = cos(2π/3) = -1/2. This means that the x-coordinate of the point is -1/2. The sine of -16π/3 is given by sin(-16π/3) = sin(2π/3) = √3/2. This means that the y-coordinate of the point is √3/2.
Therefore, the ordered pair (x, y) on the unit circle corresponding to the real number r = -16π/3 is (-1/2, √3/2).
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Suppose only the top 20 % of marks on a university entrance exam qualifies an application for admission. If the test results had a mean of 400 and standard deviation of 25 what is the minimum score for admission?
The minimum score required for admission in the university, considering only the top 20% of marks, can be determined using the given information. With a mean of 400 and a standard deviation of 25, the minimum score for admission is calculated to be 425.
To find the minimum score for admission, we need to determine the cutoff point that corresponds to the top 20% of marks. In a normal distribution, the mean marks would be at the 50th percentile. Since we are considering the top 20%, the cutoff point would be at the 80th percentile.
To calculate this cutoff point, we can use the concept of z-scores. A z-score measures the number of standard deviations a particular value is from the mean. In this case, we need to find the z-score that corresponds to the 80th percentile.
The 80th percentile corresponds to a z-score of approximately 0.84. Using the z-score formula, z = (x - μ) / σ, where x is the score, μ is the mean, and σ is the standard deviation, we can rearrange the formula to solve for x.
Plugging in the known values, we have 0.84 = (x - 400) / 25. Rearranging the equation and solving for x, we find x = 425.
Therefore, the minimum score required for admission is 425. This means that any applicant who scores 425 or above on the entrance exam will qualify for admission to the university.
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Answer the following question regarding the normal distribution:
Show that the density function
N = (μ, σ²)
is symmetric about
x = μ
, reaches its maximum in
x = μ
and has turning points at
x1 = μ + σ y x2 = μ - σ
The density function N = (μ, σ²) is symmetric about x = μ, reaches its maximum at x = μ.
and it has turning points at x₁ = μ + σ and x₂ = μ - σ.
To show that the density function of the normal distribution is symmetric about x = μ,
Demonstrate that the function is unchanged when reflected across x = μ.
The density function of the normal distribution is,
N(x) = (1 / √(2πσ²)) × exp(-((x - μ)² / (2σ²)))
To show symmetry, evaluate N(x) at x = μ + k, where k is any constant,
N(μ + k)
= (1 / √(2πσ²)) × exp(-((μ + k - μ)² / (2σ²)))
= (1 / √(2πσ²)) × exp(-((k²) / (2σ²)))
Next, we evaluate N(x) at x = μ - k,
N(μ - k)
= (1 / √(2πσ²)) ×exp(-((μ - k - μ)² / (2σ²)))
= (1 / √(2πσ²)) × exp(-((k²) / (2σ²)))
We can see that N(μ + k) = N(μ - k), which shows that the density function is symmetric about x = μ.
To determine where the density function reaches its maximum, find the value of x that maximizes N(x).
Taking the derivative of N(x) with respect to x and setting it equal to zero to find the critical points.
dN(x) / dx = -(x - μ) / (σ²) × N(x) = 0
Since N(x) is never zero, the critical points occur when (x - μ) = 0, which implies x = μ.
Thus, the density function reaches its maximum at x = μ.
To find the turning points, we can determine the second derivative of N(x) with respect to x,
d²N(x) / dx² = -N(x) / σ² + (x - μ)² × N(x) / (σ⁴)
Setting this second derivative equal to zero,
-N(x) / σ² + (x - μ)² × N(x) / (σ⁴) = 0
Simplifying the equation,
(x - μ)² = σ²
Taking the square root of both sides,
x - μ = ±σ
Solving for x,
x₁= μ + σ
x₂ = μ - σ
The turning points occur at x₁ = μ + σ and x₂ = μ - σ.
Therefore, density function of normal distribution is symmetric about x = μ, reaches its maximum at x = μ, and has turning points at x₁ = μ + σ and x₂ = μ - σ.
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You invested $25,000 in two accounts paying 4% and 7% annual interest, respectively. If the total interest earned for the year was $1480, how much was invested at each rate?
The amount invested at 4% is __
The total interest earned for the year was $1480, The amount invested at 4% is $10,000.
Let's assume the amount invested at 4% is x dollars. Since the total investment is $25,000, the amount invested at 7% would be (25,000 - x) dollars.
To calculate the total interest earned for the year, we can use the formula: Total Interest = Interest from 4% Account + Interest from 7% Account.
The interest earned from the 4% account is given by: 0.04x.
The interest earned from the 7% account is given by: 0.07(25,000 - x).
According to the problem, the total interest earned is $1480. So we can set up the equation: 0.04x + 0.07(25,000 - x) = 1480
Simplifying the equation:
0.04x + 1750 - 0.07x = 1480
-0.03x = 1480 - 1750
-0.03x = -270
x = (-270)/(-0.03)
x = 9000
Therefore, the amount invested at 4% is $9,000, and the amount invested at 7% is $25,000 - $9,000 = $16,000.
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2. [5pts.] COS X = Solve - 2 sin ZA 2 √₂ 2 for X:
The solution for x of expression cos⁻¹ x = - π/2 - 2 sin⁻¹ (√2/2) is,
⇒ x = - 1
We have to given that,
Expression to solve,
⇒ cos⁻¹ x = - π/2 - 2 sin⁻¹ (√2/2)
Now, We can simplify the expression for x,
⇒ cos⁻¹ x = - π/2 - 2 sin⁻¹ (√2/2)
⇒ cos⁻¹ x = - π/2 - 2 sin⁻¹ (1/√2)
⇒ cos⁻¹ x = - π/2 - 2 sin⁻¹ (sin π/4)
⇒ cos⁻¹ x = - π/2 - 2 × π/4
⇒ cos⁻¹ x = - π/2 - π/2
⇒ cos⁻¹ x = - π
⇒ x = cos (- π)
⇒ x = cos π
⇒ x = - 1
Therefore, The solution for x of expression cos⁻¹ x = - π/2 - 2 sin⁻¹ (√2/2) is,
⇒ x = - 1
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a chain lying on the ground is 10 m long and its mass is 70 kg. how much work (in j) is required to raise one end of the chain to a height of 4 m? (use 9.8 m/s2 for g.)
Answer:
548.8 joules!
A fair 6-sided die is rolled. What is the probability that a even number is rolled? O 0.5 0.333 0 0.6 0.167
Thus, the probability of rolling an even number is 3/6 or 1/2.
There are six possible outcomes when rolling a fair six-sided die.
These outcomes are 1, 2, 3, 4, 5, and 6. Three of these outcomes are even numbers, 2, 4, and 6.
Therefore, the probability of rolling an even number is 3/6 or 1/2.
=0.5.
So let me explain to you some concepts related to probability.
When it comes to probability, the number of outcomes is the total number of possible results.
Probability is always a number between 0 and 1. The probability of an event equals the number of ways that the event can occur, divided by the total number of possible outcomes.
A fair six-sided die has six possible outcomes, each of which has the same probability of 1/6. A die can show any number from 1 to 6.
The possible outcomes of rolling a six-sided die are:
1, 2, 3, 4, 5, 6.
Three of these outcomes are even numbers: 2, 4, and 6.
Thus, the probability of rolling an even number is the number of ways that an even number can occur, divided by the total number of possible outcomes.
There are three ways to roll an even number. They are:2, 4, and 6.
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