The absolute maximum and minimum for the function f(x) = x^4 - 4x^3 - 10 are as follows: (A) on the interval [-1,1], there is no absolute maximum; (B) on the interval [0,4], the absolute maximum occurs at x = 2; (C) on the interval [-1,2], the absolute maximum occurs at x = 2.
To find the absolute maximum and minimum of the function, we need to analyze the critical points and the endpoints of the given intervals.
(A) On the interval [-1,1], we first find the critical points by taking the derivative of f(x) and setting it equal to zero: f'(x) = 4x^3 - 12x^2 = 0. Solving this equation, we get x = 0 and x = 3. However, since 3 is not within the interval [-1,1], there are no critical points in the interval. Therefore, we check the endpoints of the interval, which are f(-1) = -14 and f(1) = -12. The function does not have an absolute maximum in this interval.
(B) On the interval [0,4], we find the critical points by setting f'(x) = 0: 4x^3 - 12x^2 = 0. Solving this equation, we find x = 0 and x = 3. However, 0 is not within the interval [0,4]. Therefore, we check the endpoints: f(0) = -10 and f(4) = 26. The absolute maximum occurs at x = 2, where f(2) = 2^4 - 4(2)^3 - 10 = 2.
(C) On the interval [-1,2], we find the critical points by setting f'(x) = 0: 4x^3 - 12x^2 = 0. Solving this equation, we get x = 0 and x = 3. However, 3 is not within the interval [-1,2]. We check the endpoints: f(-1) = -14 and f(2) = -10. The absolute maximum occurs at x = 2, where f(2) = 2^4 - 4(2)^3 - 10 = 2.
Therefore, the answers are: (A) No absolute maximum, (B) Absolute maximum at x = 2, and (C) Absolute maximum at x = 2.
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use the shell method to write and evaluate the integral that gives the volume of the solid generated by revolving the plane region about the x-axis. x y2 = 36
The volume of the solid generated by revolving the plane region about the x-axis is [tex]72\pi[/tex][tex]ln(6)[/tex].
To use the shell method to write and evaluate the integral that gives the volume of the solid generated by revolving the plane region about the x-axis, x y2 = 36, we need to first sketch the graph.
The graph of the given function is given below:
[tex]\int[/tex][tex]_{0}[/tex][tex]^{6}[/tex][tex]2[/tex][tex]\pi[/tex][tex]x[/tex][tex](\frac{36}{x}) dx[/tex][tex]\Rightarrow[/tex][tex]\int[/tex][tex]_{0}[/tex][tex]^{6}[/tex][tex]72\pi[/tex][tex]\frac{1}{x}[/tex]dx[tex]\Rightarrow[/tex][tex]72\pi[/tex][tex]\int[/tex][tex]_{0}[/tex][tex]^{6}[/tex][tex]\frac{1}{x}[/tex]dx[tex]\Rightarrow[/tex][tex]72\pi[/tex][tex]ln(x)[/tex][tex]\Biggr|_{0}^{6}[/tex][tex]\Rightarrow[/tex][tex]72\pi[/tex][tex]ln(6)[/tex].
Therefore, the volume of the solid generated by revolving the plane region about the x-axis is [tex]72\pi[/tex][tex]ln(6)[/tex].
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Multiply two rotation matrices Ta and T8 to deduce the formulas for sin(a + B) and cos(a + B). Explain your reasoning.
Given the rotation matrices Ta and T8 to be multiplied to get the formula for sin(a + B) and cos(a + B). Ta and T8 are given by,
Ta = [cos a −sin a; sin a cos a]
T8 = [cos 8 −sin 8; sin 8 cos 8]
Now, the product of Ta and T8 will give us the matrix,
TM = Ta.
T8= [cos a −sin a; sin a cos a].[cos 8 −sin 8; sin 8 cos 8]
Let's multiply both matrices to get the product matrix.
TM= [cos a cos 8 − sin a sin 8 − cos a sin 8 − sin a cos 8;sin a cos 8 + cos a sin 8 cos a cos 8 − sin a sin 8]
Since the composition of rotations is associative, we can evaluate TM as the product of the rotation matrices in the opposite order,
TM= [cos 8 cos a − sin 8 sin a − cos 8 sin a − sin 8 cos a;sin 8 cos a + cos 8 sin a cos 8 − sin 8 sin a]
Now, sin (a + 8) is given by the element at position (1, 2) in the matrix TM, while cos (a + 8) is given by the element at position (1, 1) in TM.
sin (a + 8) = −cos a sin 8 − sin a cos 8
= −sin a cos 8 + cos a sin 8
= sin a cos(8) − cos a sin(8)cos (a + 8)
= cos a cos 8 − sin a sin 8
= cos 8 cos a − sin 8 sin a
Thus, the formulas for sin (a + 8) and cos (a + 8) have been deduced using the given rotation matrices Ta and T8.
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For the following population of N=8 scores: 1, 3, 1, 10, 1, 0,
1, 3
Calculate SS
Calculate σ2
Calculate σ
Question 2 options:
Thus, the standard deviation of this population is 3.0.
Mean value = (1+3+1+10+1+0+1+3)/8= 20/8= 2.5
Thus,
SS = Σ(X – M)²= (1-2.5)² + (3-2.5)² + (1-2.5)² + (10-2.5)² + (1-2.5)² + (0-2.5)² + (1-2.5)² + (3-2.5)²
= (-1.5)² + 0.5² + (-1.5)² + 7.5² + (-1.5)² + (-2.5)² + (-1.5)² + 0.5²
= 2.25 + 0.25 + 2.25 + 56.25 + 2.25 + 6.25 + 2.25 + 0.25
= 72.0
Now, to calculate σ² (variance), we can use the following formula:
σ² = SS / N= 72.0 / 8= 9.0
Therefore, we get the variance of this population as 9.0.
To calculate σ (standard deviation), we can use the following formula:σ = √(σ²)= √(9.0)= 3.0
Thus, the standard deviation of this population is 3.0.
Hence, the SS (sum of squares), variance (σ²), and standard deviation (σ) of the given population N=8 scores: 1, 3, 1, 10, 1, 0, 1, 3 are 72.0, 9.0, and 3.0 respectively.
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The phrase is: 4 divided by the sum of 4 and a number
The algebraic expression for the phrase "4 divided by the sum of 4 and a number" is written as 4/(4 + x).
To translate the phrase "4 divided by the sum of 4 and a number" into an algebraic expression, we start by representing the unknown number with a variable, such as "x." The sum of 4 and the unknown number is expressed as "4 + x." To find the division, we write "4 divided by (4 + x)," which is mathematically represented as 4/(4 + x).
This expression indicates that we are dividing the number 4 by the sum of 4 and the unknown number "x." By using algebraic notation, we can manipulate and solve equations involving this expression to find values for "x" that satisfy specific conditions or equations.
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I need these high school statistics questions to be solved. It
would be great if you write the steps on paper, too.
38. It is estimated that 13% of people in Scotland have red hair. Find the mean and standard deviation of the number of red-headed Scots in a randomly selected group of 120. A. 0.13; 120 B. 15.6; 0.01
The mean and standard deviation of the number of red-headed Scots in a randomly selected group of 120 is 15.6 and 3.7358 respectively.
Mean or expected value
μ = np = 120 × 0.13 = 15.6
The variance of the binomial distribution is σ² = npq
where q = 1 - p and n = 120
Therefore, σ² = 120 × 0.13 × 0.87 = 13.9626
The standard deviation of the binomial distribution is:
σ = √13.9626 = 3.7358
Hence, the mean and standard deviation of the number of red-headed Scots in a randomly selected group of 120 is 15.6 and 3.7358 respectively.
Option B. 15.6; 0.01 is incorrect because the correct standard deviation is 3.7358, not 0.01.
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what is the probability of 5 cards poker hand contain two diamond and 3 of the splades
To calculate the probability of a 5-card poker hand containing two diamonds and three spades, we need to consider the total number of possible 5-card hands and the number of favorable outcomes.
Total number of possible 5-card hands:
There are 52 cards in a deck, and we want to choose 5 cards. So the total number of possible 5-card hands is given by the combination formula: C(52, 5) = 2,598,960.
Number of favorable outcomes:
We want exactly two diamonds and three spades. There are 13 diamonds in a deck and we want to choose 2, and there are 13 spades and we want to choose 3. So the number of favorable outcomes is given by: C(13, 2) * C(13, 3) = 78 * 286 = 22,308.
Probability:
The probability is calculated by dividing the number of favorable outcomes by the total number of possible outcomes:
Probability = Number of favorable outcomes / Total number of possible outcomes
Probability = 22,308 / 2,598,960 ≈ 0.0086
Therefore, the probability of a 5-card poker hand containing exactly two diamonds and three spades is approximately 0.0086 or 0.86%.
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Assume that random guesses are made on a 5 multiple choice ACT
test, so there is n=5 trials, with the probability of correct given
by p=0.20 use binomial probability
A) Find the probability that the n
The probability that the n = 5 guesses are all incorrect using binomial probability is 0.32768. Given that random guesses are made on a 5 multiple choice ACT test, there are n = 5 trials, with the probability of correct given by p = 0.20.
We have to find the probability that the n = 5 guesses are all incorrect using binomial probability. The binomial probability is used to find the probability of the x number of successes in n independent trials.
The formula for binomial probability is :P(x) = ([tex]nCx[/tex]) * [tex]p^x[/tex]* [tex]q^(n-x)[/tex] where [tex]nCx = n! / (x! * (n-x)!)[/tex] and q = 1 - p.
To find the probability that the n = 5 guesses are all incorrect, we need to find the probability that the x = 0 guesses are correct. So, we have: x = 0, n = 5, p = 0.20,
q = 1 - p
= 0.80P(x = 0)
= 5C₀ * 0.20⁰ * 0.80⁵
= 1 * 1 * 0.32768
= 0.32768
Therefore, the probability that the n = 5 guesses are all incorrect using binomial probability is 0.32768.
Answer: The probability that the n = 5 guesses are all incorrect using binomial probability is 0.32768.
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A95% confidence interval for a proportion is 0.74 to 0.83. Is the value given a plausible value of p? (a) p = 091 (b) p = 0.75 (c) p = 0.13
The only plausible value of p from the given options is p = 0.75.
We are given a 95% confidence interval for a proportion as 0.74 to 0.83. We need to determine if the given value is a plausible value of p. We can do this by finding the point estimate for the proportion using the midpoint of the confidence interval.
The midpoint of the confidence interval is given as:
Midpoint of confidence interval = (0.74 + 0.83)/2 = 0.785
This is the point estimate for the proportion p. Now we need to check if the given value is plausible or not.(a) p = 0.91 is not plausible because it is greater than the upper limit of the confidence interval.
(b) p = 0.75 is plausible because it is close to the point estimate of 0.785.(c) p = 0.13 is not plausible because it is less than the lower limit of the confidence interval.
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Score on last try: 0 of 1 pts. See Details for more. > Next question For a standard normal distribution, find: P(-1.84 <2<2.69) Question Help: Video 1 Video 2 Message Instructor Submit Question Jump to Answer Get a similar question You can retry this question below D
For a standard normal distribution, we are required to find P(-1.84 < 2 < 2.69).Solution:According to the standard normal distribution, the mean is 0 and the standard deviation is 1.
The standard normal distribution can be converted to a standard normal distribution by making the following transformation:z = (x-μ)/σ, where μ is the mean and σ is the standard deviation.The given values are: lower limit = -1.84 and upper limit = 2.69.z1 = (-1.84-0)/1 = -1.84z2 = (2.69-0)/1 = 2.69The values of z for the lower and upper limits are -1.84 and 2.69, respectively. Thus, P(-1.84 < z < 2.69) needs to be determined.Using the standard normal table, we find that P(-1.84 < z < 2.69) is equal to 0.9964. Therefore, the probability that z lies between -1.84 and 2.69 is 0.9964 or 99.64%.The standard normal table is the standard normal distribution's table of values. It helps to find the probabilities of the given values in the standard normal distribution, where the mean is 0 and the standard deviation is 1.
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Use a Maclaurin series in the table below to obtain the Maclaurin series for the given function. X) 4x2 tan 1 (3x3 SC R 1 1 x n-0 1 00 2! 3! n-o n (-1)" Sin (2n 1)! 3! 5! 7! cos X (-1) (2n)! 2! 6! n-0 2n+ 1 (-1) R 1 tan 2n 1 k(km k(k 1)(k 1 2! 3!
Maclaurin series:Maclaurin series can be defined as a power series that is a Taylor series approximation for a function at 0. Maclaurin series is a special case of the Taylor series, where a = 0. The formula for the Maclaurin series is: f(x) = f(0) + f′(0)x + f′′(0)x²/2! + f‴(0)x³/3! + …Here, we have given a table which contains Maclaurin series of different functions.
We need to use a Maclaurin series in the table to obtain the Maclaurin series for the given function. X) 4x² tan 1 (3x³)SC R 1 1 x n-0 1 00 2! 3! n-o n (-1)" Sin (2n 1)! 3! 5! 7! cos X (-1) (2n)! 2! 6! n-0 2n+ 1 (-1) R 1 tan 2n 1 k(km k(k 1)(k 1 2! 3!Given function is: 4x²tan(3x³)The formula for Maclaurin series of tan(x) is given as: tan(x) = x - x³/3 + 2x⁵/15 - 17x⁷/315 + …Using this formula, we get: tan(3x³) = 3x³ - (3x³)³/3 + 2(3x³)⁵/15 - 17(3x³)⁷/315 + …= 3x³ - 3x⁹/3 + 54x¹⁵/15 - 4913x²¹/315 + …= 3x³ - x⁹ + 18x¹⁵ - 4913x²¹/315 + …Putting this value in the given function,
we get: 4x²tan(3x³) = 4x²[3x³ - x⁹ + 18x¹⁵ - 4913x²¹/315 + …] = 12x⁵ - 4x¹¹ + 72x¹⁷ - 4913x²³/315 + …Hence, the required Maclaurin series for the given function is 12x⁵ - 4x¹¹ + 72x¹⁷ - 4913x²³/315 + …. The word count of the answer is 129 words.
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Express tan(pi/4-x) in its simplest form. Show work.
tan(pi /4-×)=(tan45-tanx)/1+tan45.tanx
=(1-tanx)/1+tanx
Here is a bivariate data set. X y 77 32.8 53.1 72.7 78.6 30.9 49.3 58.4 86.7 14.3 Find the correlation coefficient and report it accurate to three decimal places. r = Submit Question
The correlation coefficient of this bivariate data set is -0.951.
How to find an equation of the line of best fit and the correlation coefficient?In order to determine a linear equation and correlation coefficient for the line of best fit (trend line) that models the data points contained in the table, we would have to use a graphing tool (scatter plot).
In this scenario, the x-values would be plotted on the x-axis of the scatter plot while the y-values would be plotted on the y-axis of the scatter plot.
From the scatter plot (see attachment) which models the relationship between the x-values and y-values, a linear equation for the line of best fit and correlation coefficient are as follows:
Equation: y = 133.82 - 1.34x
Correlation coefficient, r = -0.950977772 ≈ -0.951.
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Please show work clearly and graph.
2. A report claims that 65% of full-time college students are employed while attending college. A recent survey of 110 full-time students at a state university found that 80 were employed. Use a 0.10
1. Null Hypothesis (H0): The proportion of employed students is equal to 65%.
Alternative Hypothesis (HA): The proportion of employed students is not equal to 65%.
2. We can use the z-test for proportions to test these hypotheses. The test statistic formula is:
[tex]\[ z = \frac{{p - p_0}}{{\sqrt{\frac{{p_0(1-p_0)}}{n}}}} \][/tex]
where:
- p is the observed proportion
- p0 is the claimed proportion under the null hypothesis
- n is the sample size
3. Given the data, we have:
- p = 80/110 = 0.7273 (observed proportion)
- p0 = 0.65 (claimed proportion under null hypothesis)
- n = 110 (sample size)
4. Calculating the test statistic:
[tex]\[ z = \frac{{0.7273 - 0.65}}{{\sqrt{\frac{{0.65 \cdot (1-0.65)}}{110}}}} \][/tex]
[tex]\[ z \approx \frac{{0.0773}}{{\sqrt{\frac{{0.65 \cdot 0.35}}{110}}}} \][/tex]
[tex]\[ z \approx \frac{{0.0773}}{{\sqrt{\frac{{0.2275}}{110}}}} \][/tex]
[tex]\[ z \approx \frac{{0.0773}}{{0.01512}} \][/tex]
[tex]\[ z \approx 5.11 \][/tex]
5. The critical z-value for a two-tailed test at a 10% significance level is approximately ±1.645.
6. Since our calculated z-value of 5.11 is greater than the critical z-value of 1.645, we reject the null hypothesis. This means that the observed proportion of employed students differs significantly from the claimed proportion of 65% at a 10% significance level.
7. Graphically, the critical region can be represented as follows:
[tex]\[ | | \\ | | \\ | \text{Critical} | \\ | \text{Region} | \\ | | \\ -------|---------------------|------- \\ -1.645 1.645 \\ \][/tex]
The calculated z-value of 5.11 falls far into the critical region, indicating a significant difference between the observed proportion and the claimed proportion.
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You wish to test the following claim ( H
a
) at a significance level of
α
=
0.05
.
H
o
:
μ
=
70.7
H
a
:
μ
≠
70.7
You believe the population is normally distributed and you know the standard deviation is
σ
=
13.5
. You obtain a sample mean of
M
=
64.1
for a sample of size
n
=
26
.
What is the test statistic for this sample? (Report answer accurate to three decimal places.)
test statistic = What is the p-value for this sample? (Report answer accurate to four decimal places.)
p-value =
The test statistic is given as follows:
[tex]z = \frac{\overline{x} - \mu}{\frac{\sigma}{\sqrt{n}}}[/tex]
In which:
[tex]\overline{x}[/tex] is the sample mean.[tex]\mu[/tex] is the value tested at the null hypothesis.[tex]\sigma[/tex] is the standard deviation of the population.n is the sample size.The parameters for this problem are given as follows:
[tex]\overline{x} = 64.1, \mu = 70.7, n = 26, \sigma = 13.5[/tex]
Hence the test statistic is given as follows:
[tex]z = \frac{64.1 - 70.7}{\frac{13.5}{\sqrt{26}}}[/tex]
z = -2.49.
Using a z-distribution calculator, considering a two tailed test, the p-value is given as follows:
0.0128.
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In a random sample of 19 people, the mean commute time to work was 30.4 minutes and the standard deviation was 7.2 minutes. Assume the population is normally distributed and use a t-distribution to construct a 95% confidence interval for the population mean u. What is the margin of error of u? Interpret the results. ... The confidence interval for the population mean u is (26.9.33.9) (Round to one decimal place as needed.) The margin of error of μ is (Round to one decimal place as needed.)
The margin of error for the population mean is approximately 3.475 minutes.
To calculate the margin of error for the population mean, we can use the formula:
Margin of Error = Critical Value * Standard Error
The critical value for a 95% confidence interval with a sample size of 19 can be obtained from the t-distribution table. The degrees of freedom for this calculation would be n - 1 = 18.
Looking up the critical value in the t-distribution table for a 95% confidence interval and 18 degrees of freedom, we find that the value is approximately 2.101.
The standard error can be calculated by dividing the standard deviation by the square root of the sample size:
Standard Error = Standard Deviation / √(Sample Size)
Plugging in the values, we get:
Standard Error = 7.2 / √(19) ≈ 1.653
Now we can calculate the margin of error:
Margin of Error = 2.101 * 1.653 ≈ 3.475
Therefore, the margin of error for the population mean is approximately 3.475 minutes.
Interpretation:
The 95% confidence interval for the population mean commute time is (26.9, 33.9) minutes. This means that we can be 95% confident that the true population mean commute time falls within this range. Additionally, the margin of error of 3.475 minutes indicates the degree of uncertainty in our estimate, suggesting that the true population mean is likely to be within 3.475 minutes of the sample mean of 30.4 minutes.
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A math class has 12 students. There are 6 tables in the classroom with exactly 2 students per table. To prevent excessive copying on a certain upcoming quiz, the math professor makes 3 different versions of the quiz with four of each of the three versions. The math professor then shuffles the quizzes and distributes them at random to the students in the class. (a) What is the probability that none of the tables have two of the same version of the quiz? (b) Define a set of tables T = {T₁, T2, T3, T4, T5, T6) Define a sample space S = { all ways to distribute two versions of the quiz to each table T, € T} Define a Bernoulli random variable for each s € S by Jo no tables in s have two of the same version X(s) = at least one table in s has two of the same version Find the probability mass function (pmf) for X. Hint P(X= 0) = the correct answer to part (a). (c) Sketch a graph of the cumulative distribution function (cdf) for X below.
To calculate the probability that none of the tables have two of the same version of the quiz, we can use the permutation formula: 4*3*2=24 ways to distribute the quizzes to the students in the class randomly. We can start by calculating the number of ways to distribute the quizzes so that each table has different quizzes.
To do that, we'll use the following formula for permutations:
6! (4!2!2!)^6. For each table, there are 4! ways to distribute the quizzes among the two students and 2! ways to arrange the quizzes for each student.
There are six tables, so multiply this by (4!2!2!)^6. The denominator is the total number of possible permutations, which is 3^12. Therefore, the probability is:
6!(4!2!2!)^6/3^12
=0.01736
(b) Let's define the set of tables T = {T₁, T2, T3, T4, T5, T6} and the sample space S = {all ways to distribute two versions of the quiz to each table T, € T}. Then, we can define a Bernoulli random variable for each s € S as follows: X(s) = 0, if no tables in s have two of the same version X(s), if at least one table in s has two of the same version find the probability mass function (pmf) for X, we can count the number of ways to distribute the quizzes for each value of X(s, and divide by the total number of possible outcomes.
P(X=0) is the probability that none of the tables have two of the same version of the quiz, which we calculated in part (a) as 0.01736.
P(X=1) is the complement of P(X=0), which is
1 - P(X=0)
= 0.98264.
(c)To sketch a graph of the cumulative distribution function (cdf) for X, we need to calculate the cumulative probabilities for each value of X. The cdf for X is defined as:
F(x) = P(X ≤ x)
For X=0, the cumulative probability is simply
P(X=0) = 0.01736.
For X=1, the cumulative probability is
F(1) = P(X ≤ 1)
= P(X=0) + P(X=1)
= 0.01736 + 0.98264
= 1.0
Therefore, the graph of the cdf for X is shown below. The probability that none of the tables have two of the same version of the quiz is 0.01736. To find the probability mass function (pmf) for the Bernoulli random variable X, we counted the number of ways to distribute the quizzes for each value of X(s). We divided by the total number of possible outcomes.
We found that P(X=0) = 0.01736 and P(X=1) = 0.98264. Finally, we sketched the graph of the cumulative distribution function (cdf) for X, which shows that the probability of having at least one table with two of the same version of the quiz increases as the number of tables increases.
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Problem 2.Suppose we are researchers at the Galapagos Tortoise Rescarch Center, and we are watching 3 tortoise eggs,waiting to record the vital statistics of the newly hatched tortoises. There is a 60% chance any of the eggs will hatch into a female tortoise and a 40% chanoe it will hatch into a male tortoise. The sex of every egg is independent of the others a. From the thrce tortoise eggs,what is the probability of getting at least one male tortoise? tortoises? c. From the three tortoise eggs,what is the probability of getting exactly 2 male tortoises? d. From the three tortoise eggs,what is the probability of getting either 1 or 3 female tortoises?
There is a 60% chance any of the eggs will hatch into a female tortoise and a 40% chanoe it will hatch into a male tortoise. The probability of getting at least one male tortoise from the three tortoise eggs is 88.8%, that ofgetting at least one male tortoise is 1 - 0.216 = 0.784 or 78.4%.
To calculate this probability, we can use the concept of complementary probability. The complementary probability of an event A is equal to 1 minus the probability of the event not happening (A'). In this case, the event A represents getting at least one male tortoise.
The probability of getting no male tortoise from a single egg is 0.6 (the probability of hatching a female tortoise). Since the sex of each egg is independent of the others, the probability of getting no male tortoise from all three eggs is 0.6 * 0.6 * 0.6 = 0.216.
Therefore, the probability of getting at least one male tortoise is 1 - 0.216 = 0.784 or 78.4%.
The probability of getting exactly 2 male tortoises from the three tortoise eggs is 43.2%.
To calculate this probability, we can use the concept of combinations. The number of ways to choose 2 out of 3 eggs to be male is given by the combination formula C(3, 2) = 3.
Additionally, we need to consider the probabilities of getting male tortoises for those 2 chosen eggs (0.4 * 0.4 = 0.16) and the probability of getting a female tortoise for the remaining egg (0.6).
Multiplying these probabilities together, we get 3 * 0.16 * 0.6 = 0.288.
Therefore, the probability of getting exactly 2 male tortoises is 0.288 or 28.8%.
The probability of getting either 1 or 3 female tortoises from the three tortoise eggs is 86.4%.
To calculate this probability, we can use the concept of combinations. The number of ways to choose 1 out of 3 eggs to be female is given by the combination formula C(3, 1) = 3.
Similarly, the number of ways to choose 3 out of 3 eggs to be female is C(3, 3) = 1. For each of these cases, we need to consider the probabilities of getting female tortoises for the chosen eggs (0.6 * 0.4 * 0.4 = 0.096) and the probability of getting a male tortoise for the remaining eggs (0.4).
Multiplying these probabilities together and summing up the results, we get 3 * 0.096 * 0.4 + 1 * 0.4 = 0.2592 + 0.4 = 0.6592.
Therefore, the probability of getting either 1 or 3 female tortoises is 0.6592 or 65.92%.
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evaluate the indefinite integral. (use c for the constant of integration.) (8t 5)2.7 dt
Given the indefinite integral as[tex]`(8t^5)^(2.7) dt`[/tex]. Let us evaluate it now. Indefinite integral is represented by [tex]`∫f(x)dx`[/tex]. It is the reverse of the derivative. Here, we need to find the primitive function that has [tex]`(8t^5)^(2.7) dt`[/tex]as its derivative. We use the formula for integration by substitution: [tex]∫f(g(x))g′(x)dx=∫f(u)du.[/tex]
Here, the given function is [tex]`f(t) = (8t^5)^(2.7)`[/tex]. Let[tex]`u = 8t^5`.[/tex] Now, [tex]`du/dt = 40t^4`.⇒ `dt = du/40t^4`.[/tex] Hence, the indefinite integral [tex]`(8t^5)^(2.7) dt`[/tex]becomes,[tex]`∫(8t^5)^(2.7) dt``= ∫u^(2.7) du/40t^4`[/tex] (Substituting [tex]`u = 8t^5`[/tex]) `= (1/40) [tex]∫u^(2.7)/t^4 du` `= (1/40) ∫(u/t^4)^(2.7) du` `= (1/40) [(u/t^4)^(2.7+1)/(2.7+1)] + c` `= (1/40) [(8t^5/t^4)^(2.7+1)/(2.7+1)] + c` `= (1/40) [(8t)^(13.5)/(13.5)] + c` `= (1/540) [(8t)^(13.5)] + c`[/tex]
Therefore, the indefinite integral [tex]`(8t^5)^(2.7) dt`[/tex]is [tex]`(1/540) [(8t)^(13.5)] + c`[/tex]. Hence, the solution is [tex]`(1/540) [(8t)^(13.5)] + c`[/tex]where [tex]`c`[/tex] is a constant of integration.
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Jamie needs to multiply 2x-4 and 2x^2 + 3xy -2y^2 they decided to use the box method fill the spaces in the table with the products when multiplying each term
Answer:
2x^2 | 3xy | -2y^2
--------------------------------------
2x | 4x^3 6(x^2)y -4x(y^2)
-4 | -8x^2 -12xy 8y^2
find the taylor series representation of f(x) = cos x centered at x = pi/2
The Taylor series representation of f(x) = cos x centered at x = pi/2 is - (x - pi/2) + (1/6)(x - pi/2)^3 + .The Taylor series representation of f(x) = cos x centered at x = pi/2 is - (x - pi/2) + (1/6)(x - pi/2)^3 + ...
To find the Taylor series representation of f(x) = cos x centered at x = pi/2, we will follow these steps: Step 1: Find the value of f(x) and its derivatives at x = pi/2Step 2: Write out the general form of the Taylor series Step 3: Substitute the values of the function and its derivatives into the general form of the Taylor series Step 4: Simplify the resulting series by combining like terms.
Let's begin with step 1:Find the value of f(x) and its derivatives at x = pi/2f(x) = cos x f(pi/2) = cos(pi/2) = 0f '(x) = -sin x f '(pi/2) = -sin(pi/2) = -1f ''(x) = -cos x f ''(pi/2) = -cos(pi/2) = 0f '''(x) = sin x f '''(pi/2) = sin(pi/2) = 1f ''''(x) = cos x f ''''(pi/2) = cos(pi/2) = 0
Step 2: Write out the general form of the Taylor series . The general form of the Taylor series centered at x = pi/2 is:f(x) = f(pi/2) + f '(pi/2)(x - pi/2) + (f ''(pi/2)/2!)(x - pi/2)^2 + (f '''(pi/2)/3!)(x - pi/2)^3 + (f ''''(pi/2)/4!)(x - pi/2)^4 + ...
Step 3: Substitute the values of the function and its derivatives into the general form of the Taylor seriesf(x) = 0 + (-1)(x - pi/2) + (0/2!)(x - pi/2)^2 + (1/3!)(x - pi/2)^3 + (0/4!)(x - pi/2)^4 + ...f(x) = - (x - pi/2) + (1/6)(x - pi/2)^3 + ...
Step 4: Simplify the resulting series by combining like terms .
Therefore, the Taylor series representation of f(x) = cos x centered at x = pi/2 is - (x - pi/2) + (1/6)(x - pi/2)^3 + .The Taylor series representation of f(x) = cos x centered at x = pi/2 is - (x - pi/2) + (1/6)(x - pi/2)^3 + ...
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the p-value of the test is .0202. what is the conclusion of the test at =.05?
Given that your p-value (0.0202) is less than the significance level of 0.05, we would reject the null hypothesis at the 0.05 significance level. This suggests that the observed data provides sufficient evidence to conclude that there is a statistically significant effect or relationship, depending on the context of the test.
In statistical hypothesis testing, the p-value is used to determine the strength of evidence against the null hypothesis. The p-value represents the probability of obtaining a test statistic as extreme as the one observed, assuming the null hypothesis is true.
In your case, the p-value of the test is 0.0202. When comparing this p-value to the significance level (also known as the alpha level), which is typically set at 0.05 (or 5%), the conclusion can be drawn as follows:
If the p-value is less than or equal to the significance level (p ≤ α), we reject the null hypothesis.
If the p-value is greater than the significance level (p > α), we fail to reject the null hypothesis.
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find the radius of convergence, r, of the series. [infinity] (−1)n (x − 6)n 5n 1 n = 0 r = find the interval, i, of convergence of the series. (enter your answer using interval notation.) i =
The series converges at [tex]$x = 0$[/tex].
Therefore, the interval of convergence is [tex]$i = [0, 6]$[/tex].
The series is
[tex][infinity] (−1)n (x − 6)n 5n 1 n = 0.[/tex]
We need to find the radius of convergence, r, and the interval, i, of convergence of the series.
The radius of convergence is given by:
[tex]$$r = \frac{1}{\limsup_{n\to\infty}\sqrt[n]{|a_n|}}$$[/tex]
where $a_n$ are the coefficients of the series.
Here,
[tex]$a_n = 5n$, so$$r = \frac{1}{\limsup_{n\to\infty}\sqrt[n]{|5n|}}=\frac{1}{\limsup_{n\to\infty}\sqrt[n]{5}\sqrt[n]{n}}= \frac{1}{\infty} = 0$$[/tex]
So, the radius of convergence is 0.
To find the interval of convergence, we need to check the convergence of the series at the end points of the interval,
[tex]$x = 6$[/tex] and [tex]$x = 0$.[/tex]
For [tex]$x = 6$[/tex], the series becomes:
[tex]$$\sum_{n=0}^\infty (-1)^n (6-6)^n (5n) = \sum_{n=0}^\infty 0 = 0$$[/tex]
So, the series converges at [tex]$x = 6$[/tex] .For [tex]$x = 0$[/tex], the series becomes:
[tex]$$\sum_{n=0}^\infty (-1)^n (0-6)^n (5n) = \sum_{n=0}^\infty (-1)^n (5n)$$[/tex]
This is an alternating series that satisfies the conditions of the Alternating Series Test.
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The series converges for all x, the interval of convergence is (-∞, ∞), which can be expressed in interval notation as i = (-∞, ∞).
To find the radius of convergence, we can use the ratio test. The ratio test states that for a power series
∑(a_n * (x - c)^n), if the limit of |a_(n+1) / a_n| as n approaches infinity exists, then the series converges if the limit is less than 1 and diverges if the limit is greater than 1.
In this case, we have the series ∑((-1)^n * (x - 6)^n * 5^n / n), where c = 6.
Applying the ratio test:
lim(n→∞) |((-1)^(n+1) * (x - 6)^(n+1) * 5^(n+1) / (n+1)) / ((-1)^n * (x - 6)^n * 5^n / n)|
Simplifying, we get:
lim(n→∞) |(-1) * (x - 6) * 5 / (n+1)|
Taking the absolute value and bringing constants outside the limit:
|-5(x - 6)| * lim(n→∞) (1 / (n+1))
Since lim(n→∞) (1 / (n+1)) = 0, the limit becomes:
|-5(x - 6)| * 0 = 0
For the series to converge, we need this limit to be less than 1. However, in this case, the limit is always 0 regardless of the value of x. This means that the series converges for all x, which implies that the radius of convergence, r, is infinity.
Now, let's find the interval of convergence, i. Since the series converges for all x, the interval of convergence is (-∞, ∞), which can be expressed in interval notation as i = (-∞, ∞).
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Assume that you have a sample of n₁ = 6, with the sample mean X₁ = 42, and a sample standard deviation of S, = 6, and you have an independent sample of n₂ = 8 from another population with a samp
At the 0.01 level of significance, there is no evidence that μ₁ > μ₂. Hence, the answer is no.
Assuming that the population variances are equal, at the 0.01 level of significance, whether there is evidence that μ₁ > μ₂ is to be determined.
Sample 1:
Sample size n₁ = 6,
Sample mean [tex]\bar{X_1}=42[/tex],
Sample standard deviation S₁ = 6
Sample 2:
Sample size n₂ = 8 ,
Sample mean [tex]\bar{X_2}=37[/tex],
Sample standard deviation S₂ = 5
The null hypothesis is H₀: μ₁ ≤ μ₂
The alternate hypothesis is H₁: μ₁ > μ₂
The significance level is α = 0.01
degrees of freedom = n₁ + n₂ – 2 = 6 + 8 – 2 = 12
We know that the two samples are independent and that the population variances are equal. We can now use the pooled t-test to test the hypothesis.
Assuming that the population variances are equal, the pooled t-test statistic is calculated as follows:
[tex]t = \frac{\left(\bar{X_1} - \bar{X_2}\right)}{S_p\sqrt{\frac{1}{n_1} + \frac{1}{n_2}}}[/tex]
Where Sp is the pooled standard deviation.
The formula for the pooled standard deviation is:
[tex]S_p = \sqrt{\frac{\left(n_1 - 1\right)S_1^2 + \left(n_2 - 1\right)S_2^2}{n_1 + n_2 - 2}}[/tex]
Substituting the given values, we have:
[tex]S_p = \sqrt{\frac{\left(6 - 1\right)6^2 + \left(8 - 1\right)5^2}{6 + 8 - 2}} = 5.3026[/tex]
Substituting these values in the equation for t, we have:
[tex]t = \frac{\left(42 - 37\right)}{5.3026\sqrt{\frac{1}{6} + \frac{1}{8}}}t = 2.3979[/tex]
The critical value of t for a one-tailed test with 12 degrees of freedom and α = 0.01 is:
[tex]t_{0.01,12} = 2.718[/tex]
Since the calculated value of t (2.3979) is less than the critical value of t (2.718), we do not have enough evidence to reject the null hypothesis (H₀: μ₁ ≤ μ₂).
Therefore, at the 0.01 level of significance, there is no evidence that μ₁ > μ₂. Hence, the answer is no.
The question should be:
Assume that you have a sample of n₁ = 6, with the sample mean [tex]\bar{X_1}=42[/tex], and a sample standard deviation of S₁ = 6, and you have an independent sample of n₂ = 8 from another population with a sample mean of [tex]\bar{X_2}=37[/tex] and sample standard deviation S₂ = 5. Assuming the population variances are equal , at the 0.01 level of significance ,is there evidence that μ₁ > μ₂ ?
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A reinforced concrete section beam section size b*h=250mm*500mm concrete adopts C25 reinforced adopts HRB335 bending moment design value M= 125Kn-m try to calculate the tensile reinforcement section area as and draw. the reinforcement diagram
The tensile reinforcement section area can be calculated using the formula (M * [tex]10^6[/tex]) / (0.87 * fy * d).Tensile reinforcement section area: 276.34 mm².
What is the tensile reinforcement area?To calculate the tensile reinforcement section area for the given reinforced concrete beam, we can use the following steps:
Determine the maximum allowable stress for the steel reinforcement based on the grade of steel (HRB335). The allowable stress for HRB335 is typically around 335 MPa.Calculate the required tensile reinforcement area using the formula:As = (M * [tex]10^6[/tex]) / (0.87 * fy * d)
Where:
M is the bending moment (125 kN-m in this case).
fy is the yield strength of the steel reinforcement (typically 335 MPa).
d is the effective depth of the beam, which can be taken as the total depth of the beam minus the cover.
Determine the effective depth of the beam. In this case, the total depth of the beam is 500 mm, and considering a typical cover of 25 mm on each side, the effective depth would be 500 mm - 2 * 25 mm = 450 mm.Substitute the values into the formula to calculate the required tensile reinforcement area.Using these steps, the tensile reinforcement section area can be determined, and a reinforcement diagram can be drawn accordingly. However, since I can't draw diagrams directly, I can provide the calculated value for the tensile reinforcement section area, which you can use to create the diagram.
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find the relative frequency for the class with lower class limit 27 relative frequency =?
Ages Number of students
15 - 18 3
19 - 22 3
23 - 26 9
27 - 30 5
31 - 34 8 25 - 38 8
the relative frequency for the class with lower class limit 27 is 14.29%.Hence, option (4) is the correct answer.
Given,Ages Number of students15 - 18 319 - 22 323 - 26 927 - 30 531 - 34 825 - 38 8We need to find the relative frequency for the class with lower class limit 27.ClassIntervalFrequency15-18319-22323-26927-30531-34825-38 From the given data, we have;Lower limit Upper limit Frequency Relative frequency(Percentage)15 18 3 3/35 × 100 = 60/7 ≈ 8.5719 22 3 3/35 × 100 = 60/7 ≈ 8.5723 26 9 9/35 × 100 = 180/7 ≈ 25.7127 30 5 5/35 × 100 = 100/7 ≈ 14.2931 34 8 8/35 × 100 = 160/7 ≈ 22.8635 38 8 8/35 × 100 = 160/7 ≈ 22.86Therefore, the relative frequency for the class with lower class limit 27 is 14.29%.Hence, option (4) is the correct answer.
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Question 4 (1 point) In how many ways can 4 girls and 3 boys be arranged in a row, such that all 3 boys are not sitting together?
To calculate the number of ways the 4 girls and 3 boys can be arranged in a row such that all 3 boys are not sitting together, we need to subtract the number of arrangements where the boys are sitting together from the total number of arrangements.
Total number of arrangements:
Since we have 7 individuals (4 girls and 3 boys), the total number of arrangements without any restrictions is 7!.
Number of arrangements where the boys are sitting together:
If we consider the 3 boys as a single entity, we have 5 entities to arrange (4 girls + 1 group of boys). The number of arrangements with the boys sitting together is 5!.
To find the number of arrangements where the boys are not sitting together, we subtract the number of arrangements where the boys are sitting together from the total number of arrangements:
Number of arrangements = Total number of arrangements - Number of arrangements where boys are sitting together
= 7! - 5!
Now let's calculate the values:
Total number of arrangements = 7!
= 7 x 6 x 5 x 4 x 3 x 2 x 1
= 5040
Number of arrangements where boys are sitting together = 5!
= 5 x 4 x 3 x 2 x 1
= 120
Number of arrangements where boys are not sitting together.
= 5040 - 120
= 4920
There are 4920 ways to arrange 4 girls and 3 boys in a row such that all 3 boys are not sitting together.
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Erythromycin is a drug that has been proposed to possibly lower the risk of premature delivery. A related area of interest is its association with the incidence of side effects during pregnancy. Assume that 30% of all pregnant women complain of nausea between the 24th and 28th week of pregnancy. Furthermore, suppose that of 178 women who are taking erythromycin regularly during this period, 67 complain of nausea. Find the p-value for testing the hypothesis that incidence rate of nausea for the erythromycin group is greater than for a typical pregnant woman.
(b) At the 1% significance level, what is the conclusion of the above hypothesis test?
(A) We cannot conclude that the incidence rate of nausea for the erythromycin group is greater than
for a typical pregnant woman since the p-value is less than .02 (B) We conclude that the incidence rate of nausea for the erythromycin group is greater than
for a typical pregnant woman since the p-value is less than 0.01 (C) We conclude that the incidence rate of nausea for the erythromycin group is greater than
for a typical pregnant woman since the p-value is greater than or equal to .02 (D) We cannot conclude that the incidence rate of nausea for the erythromycin group is greater than
for a typical pregnant woman since the p-value is less than 0.01 (E) We cannot conclude that the incidence rate of nausea for the erythromycin group is greater than
for a typical pregnant woman since the p-value is greater or equal to 0.01 (F) We conclude that the incidence rate of nausea for the erythromycin group is greater than
for a typical pregnant woman since the p-value is greater than or equal to 0.01 (G) We conclude that the incidence rate of nausea for the erythromycin group is greater than
for a typical pregnant woman since the p-value is less than .02 (H) We cannot conclude that the incidence rate of nausea for the erythromycin group is greater than
for a typical pregnant woman since the p-value is greater or equal to .02
The answer is (D) We cannot conclude that the incidence rate of nausea for the erythromycin group is greater than for a typical pregnant woman since the p-value is less than 0.01.
The incidence rate of nausea for the erythromycin group is greater than for a typical pregnant woman.This is a one-sided hypothesis test, because we are interested in whether erythromycin use leads to more nausea, not whether it leads to more or less nausea. For this one-sided hypothesis test, we use the one-sided p-value, which is the probability that the observed outcome would have been at least as extreme as the observed outcome, if the null hypothesis is true.
We are trying to find the p-value for testing the hypothesis that incidence rate of nausea for the erythromycin group is greater than for a typical pregnant woman.The null hypothesis and the alternative hypothesis areH0: p ≤ 0.3HA: p > 0.3Where p is the proportion of pregnant women on erythromycin who complain of nausea. Here, the null hypothesis is that erythromycin does not increase the likelihood of nausea, and the alternative hypothesis is that erythromycin increases the likelihood of nausea.
We can find the p-value for this test as follows:We will use the normal approximation to the binomial distribution, since the sample size is large and np and n(1-p) are both greater than or equal to 5, where n is the sample size and p is the probability of success. Here, n = 178 and p = 67/178 = 0.377. Therefore, np = 67 and n(1-p) = 111.We find the test statistic, which is the z-score of the sample proportion.z = (p - P) / sqrt(P(1 - P) / n)where P = 0.3 is the hypothesized proportion of pregnant women who complain of nausea without erythromycin use. We havez = (0.377 - 0.3) / sqrt(0.3 * 0.7 / 178) = 2.149We find the one-sided p-value as P(Z > 2.149) = 0.0155.
Therefore, the answer is (A) We cannot conclude that the incidence rate of nausea for the erythromycin group is greater than for a typical pregnant woman since the p-value is less than .02At the 1% significance level, the conclusion of the above hypothesis test is that we cannot reject the null hypothesis that erythromycin use does not increase the likelihood of nausea, since the p-value is greater than 0.01. Therefore, the answer is (D) We cannot conclude that the incidence rate of nausea for the erythromycin group is greater than for a typical pregnant woman since the p-value is less than 0.01.
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Problem # 3: (15pts) Consider two events X and Y with probabilities, P(X) = 7/15, P(XY)=1/3, and P(X/Y) = 2/3. Calculate P(Y), P(Y/X), and P(Y/X). State with reasons whether the events X and Y are dep
P(Y/X) = 5/7.
To calculate P(Y), we can use the formula for the total probability:
P(Y) = P(Y/X) * P(X) + P(Y/¬X) * P(¬X)
Since we don't have the value of P(Y/¬X), we cannot calculate P(Y) based on the given information.
To calculate P(Y/X), we can use the formula for conditional probability:
P(Y/X) = P(XY) / P(X)
Substituting the given values, we have:
P(Y/X) = (1/3) / (7/15) = (1/3) * (15/7) = 5/7
To calculate P(Y/X), we can use the formula for conditional probability:
P(Y/X) = P(XY) / P(X)
Substituting the given values, we have:
P(Y/X) = (1/3) / (7/15) = (1/3) * (15/7) = 5/7
Therefore, P(Y/X) = 5/7.
Based on the calculated probabilities, we cannot determine whether the events X and Y are dependent or independent without further information.
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The t-statistic is calculated by dividing the estimator minus its hypothesized value by the standard error of the estimator.
True or False
The statement is: False.
The t-statistic is not calculated by dividing the estimator minus its hypothesized value by the standard error of the estimator. In fact, the t-statistic is calculated by dividing the difference between the estimator and its hypothesized value by the standard error of the estimator. This subtle difference in calculation can have a significant impact on the interpretation of the t-statistic and its associated p-value.
To understand why this distinction is important, let's break down the calculation of the t-statistic. The numerator of the t-statistic represents the difference between the estimator and its hypothesized value. This difference measures how far the estimated value deviates from the hypothesized value. The denominator of the t-statistic, on the other hand, is the standard error of the estimator, which captures the variability or uncertainty associated with the estimator.
By dividing the difference between the estimator and its hypothesized value by the standard error of the estimator, we obtain a ratio that quantifies the magnitude of the difference relative to the uncertainty. This ratio is the t-statistic. It allows us to assess whether the difference between the estimator and its hypothesized value is statistically significant, meaning it is unlikely to have occurred by chance.
The t-statistic is then used in hypothesis testing, where we compare it to a critical value or calculate its associated p-value to determine the statistical significance of the difference. This helps us make inferences about the population parameters based on the sample data.
In summary, the t-statistic is not calculated by dividing the estimator minus its hypothesized value by the standard error of the estimator. Rather, it is calculated by dividing the difference between the estimator and its hypothesized value by the standard error of the estimator. Understanding this distinction is crucial for accurate interpretation of statistical tests and hypothesis testing.
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pls
help
X Incorrect. If the two legs in the following 45-45-90 triangle have length 21 inches, how long is the hypotenuse? 45° √2x Round your answer to two decimal places. 1 The hypotenuse is approximately
Work Shown:
[tex]\text{hypotenuse} = \text{leg}*\sqrt{2}\\\\\text{hypotenuse} = 21*\sqrt{2}\\\\\text{hypotenuse} \approx 29.69848480983\\\\\text{hypotenuse} \approx 29.70\\\\[/tex]
Note: This template formula works for 45-45-90 triangles only.
Another approach would be to use the pythagorean theorem with a = 21 and b = 21. Plug those into [tex]a^2+b^2 = c^2[/tex] to solve for c.
