2.6.4 practice modeling pendulums and bridges

Verify the identity for sin 4x+ sin 12x / cos 4x + cos 12x -= tan (8x) will give a value of sin (8x) / cos (8x)

Trigonometry is a branch of mathematics that deals with the study of relationships between the sides and angles of triangles. It has a wide range of applications in various fields, including engineering, physics, navigation, and astronomy.

There are six trigonometric functions: sine (sin), cosine (cos), tangent (tan), cosecant (csc), secant (sec), and cotangent (cot). These functions are defined based on the ratios of the sides of a right triangle.

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First we turn the 17.2 km into mm. To do that we turn it into 17,200 m then into 1,720,000 cm then into 17,200,000 mm. Then we just divide 17.2 million by 86 so 17,200,000÷86=200,000. so we know that the scale of the map is 1:200,000. Also pls mark as brainliest answer thx.

Answer:

Step-by-step explanation:

cos47=80/x

x=80/0.6820

x=117.3020 feet

Answer: Using the formula i = prt, we have:

i = (104292.12 - 80040) = 24252.12

p = 80040

t = 3

Substituting these values, we get:

24252.12 = 80040 * r * 3

Solving for r, we get:

r = 0.101 or 10.1%

Therefore, the interest rate is 10.1%.

Step-by-step explanation:

Answer:

C. Trimodal

Step-by-step explanation:

The mode of a data set is the value that appears most frequently.

If we list the given data set in increasing order, we get:

70, 75, 75, 75, 80, 83, 83, 83, 84, 84, 86, 86, 86, 88, 89

We can see that there are 3 values (75, 83, 86) that all have equally high levels of frequency. Therefore, this data set is trimodal, since tri- is the Latin root for 3.

Answer:

Road:

Distance to Greg's house = 3.2 + 1.7 =

4.9 miles

Time to get to Greg's house =

(4.9 mi)/(15 mi/hr) = about .327 hours = about 19.6 minutes (19 minutes, 36 seconds)

Shortcut:

Distance to Greg's house:

[tex] \sqrt{ {3.2}^{2} + {1.7}^{2} } = 3.6235 [/tex]

Time to get to Greg's house:

(3.6235 mi)/(15 mi/hr) = about .242 hours = about 14.49 minutes (14 minutes, 30 seconds)

The shortcut is about 4.9 - 3.6235 = 1.28 miles faster (about 5.11 minutes, or 5 minutes, 6 seconds faster)

Answer:

-12, -17, -204

Step-by-step explanation:

The truth table is given above for (A ⋀ B) → C.

What is the logical statement?

A logical statement, also known as a proposition or a statement of fact, is a declarative sentence that is either true or false, but not both. It is a statement that can be evaluated based on the available information or evidence to determine its truth value. In other words, a logical statement is a statement that can be either true or false, but not both.

To create a truth table for the logical statement (A ⋀ B) → C, we need to consider all possible combinations of truth values for propositions A, B, and C.

There are 2 possible truth values (true or false) for each proposition, so there are 2³ = 8 possible combinations.

We can organize these combinations into a table as follows:

| A | B | C | (A ⋀ B) | (A ⋀ B) → C |

|---|---|---|---------|-------------|

| T | T | T |    T    |      T      |

| T | T | F |    T    |      F      |

| T | F | T |    F    |      T      |

| T | F | F |    F    |      T      |

| F | T | T |    F    |      T      |

| F | T | F |    F    |      T      |

| F | F | T |    F    |      T      |

| F | F | F |    F    |      T      |

In this table, the column labeled (A ⋀ B) represents the truth value of the conjunction of A and B (i.e., A AND B), and the column labeled (A ⋀ B) → C represents the truth value of the conditional statement (A ⋀ B) → C.

The symbol "T" represents "true" and the symbol "F" represents "false".

Hence, The truth table is given above for (A ⋀ B) → C.

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Answer:

-2.5

Step-by-step explanation:

follow BIMDAS.

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The area of given trapezoid whose parallel sides are 8 inches, 16 inches & its perpendicular height is 6 inches is 72 inches².

What is a trapezoid?

Quadrilaterals or four-sided polygons with one pair of parallel sides and one pair of non-parallel sides are known as trapezoids, usually spelt trapezium. It has a perimeter and covers a certain amount of space. The bases of the trapezium are the sides that are parallel to one another. Legs or lateral sides indicates to the non-parallel sides. The altitude or perpendicular height is the separation between the parallel sides. This shape's area is equal to half of the product of its parallel sides times its height.

Given dimensions of trapezoid:

let parallel sides be denoted by a , b & height be 'h'

a= 8 inches (top side)

b= 8+4+4=16 inches (bottom side)

h= 6 inches

Area = [tex]\frac{sum of parallel sides}{2}[/tex] x height

         =[tex]\frac{(a+b)h}{2}[/tex]

         =[tex]\frac{(8+16)6}{2}[/tex]

          =[tex]\frac{24(6)}{2}[/tex]

          =24 x 3

          =72 sq. inches

The area of given trapezoid is 72 inches²

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Answer:

≈ 0.153

Step-by-step explanation:

using the rules of logarithms

log x - log y ⇔ log ([tex]\frac{x}{y}[/tex] )

[tex]log_{b}[/tex] b = 1

then

[tex]log_{c}[/tex] ( [tex]\frac{4}{c}[/tex] )

= [tex]log_{c}[/tex] 4 - [tex]log_{c}[/tex] c

= 1.153 - 1

= 0.153

For the given expression -  xº is greater than 0x as,  xº = 1 and 0x = 0.

The power or exponent of a number tells us how many times that number must be multiplied by itself in order to utilise it in a multiplication. Any integer, fraction, or decimal can serve as the basis in this situation. Moreover, the exponent might have either a positive or negative value.

According to the law, we must add the exponents to obtain the product of exponential statements that have the same base.

According to the law, we must subtract the exponents in order to divide two exponential expressions with the same base.

Given expression:

xº  = 1 (any number raised to power 0 is always 1)

0x = 0 (any value multiplied with 0 becomes 0).

Thus, xº is greater than 0x as,  xº = 1 and 0x = 0.

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In this expression, the GCF is 5. Therefore, we can factor out a 5 to get 5(x² - y²).

The difference of squares formula states that the difference of two squares can be factored into the product of two terms, one of which is the sum of the terms and the other is the difference of the terms.

When factoring polynomials, it is important to factor out the greatest common factor (GCF).

In this expression, the GCF is 5.

Therefore, we can factor out a 5 to get 5(x² - y²).

Now, we can expand the parentheses and factor out the remaining terms. Since both terms are perfect squares, we can factor them using the difference of squares formula:

5(x² - y²) = 5(x + y)(x - y)

In this case, the terms are x and y, resulting in the factorization of  5(x² - y²) into 5(x + y)(x - y).

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The expression included parentheses as 21+m/7, the division inside the parentheses takes priority and we divide 28 by 7 first before adding 21.

An expression is a mathematical phrase that combines numbers, variables, and operations, but does not have an equal sign or equation.

According to the given information:

To evaluate 21+m/7 for m=28, we substitute the value of m=28 into the expression and simplify according to the order of operations, which is:

1) Perform any calculations inside the parentheses, if there are any.

2) Evaluate any exponents or roots, from left to right.

3) Perform multiplication and division, from left to right.

4) Perform addition and subtraction, from left to right.

Applying this order of operations, we get:

21 + m/7

= 21 + 28/7 [Substitute m=28]

= 21 + 4 [Divide 28 by 7 to get 4]

= 25 [Add 21 and 4]

Therefore, 21+m/7 for m=28 equals 25.

Note that if the expression had been written without parentheses as 21 + m ÷ 7, the order of operations would have required us to perform the division before the addition, which would have given us:

21 + m ÷ 7

= 21 + 28 ÷ 7 [Substitute m=28]

= 21 + 4 [Divide 28 by 7 to get 4]

= 25 [Add 21 and 4]

However, since the expression included parentheses as 21+m/7, the division inside the parentheses takes priority and we divide 28 by 7 first before adding 21.

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The expression included parentheses as 21+m/7, the division inside the parentheses takes priority and we divide 28 by 7 first before adding 21.

What is expression?

An expression is a mathematical phrase that combines numbers, variables, and operations, but does not have an equal sign or equation.

According to the given information:

To evaluate 21+m/7 for m=28, we substitute the value of m=28 into the expression and simplify according to the order of operations, which is:

1) Perform any calculations inside the parentheses, if there are any.

2) Evaluate any exponents or roots, from left to right.

3) Perform multiplication and division, from left to right.

4) Perform addition and subtraction, from left to right.

Applying this order of operations, we get:

21 + m/7

= 21 + 28/7 [Substitute m=28]

= 21 + 4 [Divide 28 by 7 to get 4]

= 25 [Add 21 and 4]

Therefore, 21+m/7 for m=28 equals 25.

Note that if the expression had been written without parentheses as 21 + m ÷ 7, the order of operations would have required us to perform the division before the addition, which would have given us:

21 + m ÷ 7

= 21 + 28 ÷ 7 [Substitute m=28]

= 21 + 4 [Divide 28 by 7 to get 4]

= 25 [Add 21 and 4]

However, since the expression included parentheses as 21+m/7, the division inside the parentheses takes priority and we divide 28 by 7 first before adding 21.

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Using matrix form, the solution to the system of linear equations are 55/12 and 5/4

a) To write the system of equations in matrix form, we can use the coefficient matrix A and the vector of constants B:

| 15  -7 |   | V₁ |   |  60 |

| -3   3 | * | V₂ | = | -10 |

b) To find V₁ and V₂, we can use matrix algebra to solve the equation AV = B for V:

AV = B

V = A^-1 B

First, we need to find the inverse of the coefficient matrix A:

| 15  -7 |   |  3  7 |

| -3   3 | → | -5 15 | * (1/60)

So, A^-1 = (1/60) * | 3 7 | |-5 15 |

Solving for both equations

v₁ = 55/12, v₂ = 5/4

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Answer:

To check whether 40, -40, 16, or -16 is the solution of the equation 28 = d - 12, we can substitute each value into the equation and see if it results in a true statement.

When we substitute 40 into the equation, we get:

28 = 40 - 12

Simplifying, we get:

28 = 28

This is a true statement, so 40 is a solution of the equation.

When we substitute -40 into the equation, we get:

28 = -40 - 12

Simplifying, we get:

28 = -52

This is not a true statement, so -40 is not a solution of the equation.

When we substitute 16 into the equation, we get:

28 = 16 - 12

Simplifying, we get:

28 = 4

This is not a true statement, so 16 is not a solution of the equation.

When we substitute -16 into the equation, we get:

28 = -16 - 12

Simplifying, we get:

28 = -28

This is not a true statement, so -16 is not a solution of the equation.

Therefore, the only solution of the equation 28 = d - 12 is d = 40.

Answer:

29i

Step-by-step explanation:

While multiplying complex numbers we should remember that i^2=−1.

As p=2 and q=5i,

(2−5i)(p+q)i

= (2−5i)(2+5i)i

= (2×2+2×5i−5i×2−5i×5i)×i

= (4+10i−10i−25×i2)×i

= (4+10i−10i−25×(−1))×i

= (4+25)×i

=29i

Therefore , the solution of the given problem of probability comes out to be a)1/78 ,b)65/624 ,c)1/4 ,d)0 and e)12/13.

The basic goal of any considerations technique is to assess the probability that a statement is accurate or that a specific incident will occur. Chance can be represented by any number range between 0 and 1, where 0 normally indicates a percentage but 1 typically indicates the level of certainty. An illustration of probability displays how probable it is that a specific event will take place.

Here,

a.

P(Bert gets a Jack and Ernie rolls a five) = P(Bert gets a Jack) * P(Ernie rolls a five)

= (4/52) * (1/12)

= 1/78

b.

P(Bert gets a heart and Ernie rolls a number less than six) = P(Bert gets a heart) * P(Ernie rolls a number less than six)

= (13/52) * (5/12)

= 65/624

c.

P(Bert gets a face card and Ernie rolls an even number) = P(Bert gets a face card) * P(Ernie rolls an even number)

= (12/52) * (6/12)

= 1/4

d.

P(Bert gets a red card and Ernie rolls a fifteen) = 0

e.

Ernie rolls a number that is not twelve, and Bert draws a card that is not a Jack:

A regular 52-card deck contains 48 cards that are not Jacks,

so the likelihood that Bert will draw one of those cards is 48/52, or 12/13.

On a 12-sided dice with 11 possible outcomes,

Ernie rolls a non-12th-number (1, 2, 3, etc.).

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Answer:

The answer is 15:6 or 15 to 6

Step-by-step explanation:

white=15

black=6

white ratio black

15:6

The sample variance ratio, denoted by [tex]$\frac{S_x^2}{S_y^2}$[/tex], which follows an F-distribution.

The sample variance for a random sample of size (m) from a normal distribution with mean [tex]$\mu_X$[/tex] and common variance [tex]$\sigma^2$[/tex] is given by:

[tex]S_{x} ^2 =\frac{1}{m-1}\sum_{i=1}^{m}($X_i$-$\overline{X}$)^2[/tex]

where [tex]$X_i$[/tex] are the individual observations from the sample, and [tex]$\overline{X}$[/tex] is the sample mean.

Similarly, the sample variance for a random sample of size (n) from a normal distribution with mean [tex]$\mu_{Y} _$[/tex] and common variance [tex]$\sigma^2$[/tex] is given by:

[tex]S_{y} ^2 =\frac{1}{n-1}\sum_{i=1}^{n}($Y_i$-$\overline{Y}$)^2[/tex]

where [tex]$Y_i$[/tex] are the individual observations from the sample, and [tex]$\overline{Y}$[/tex] is the sample mean.

Provided that both samples have normal distributions with the same variance [tex]$\sigma^2$[/tex], the ratio of sample variances [tex]\frac{Sx^2}{Sy^2}[/tex] follows an F-distribution with degrees of freedom [tex]m-1$ and $n-1$[/tex], respectively.

Thus,

[tex]\frac{S_x^2}{S_y^2} $\sim$ $F(m-1,n-1)$[/tex]

where [tex]$\sim$[/tex] denotes "follows the distribution of"

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Answer:

Step-by-step explanation:

(a) The estimate for the difference in population means is 6.000.

(b) The standard error for the difference in population means is 3.208.

Conclusion:

Based on the given data, the 95% confidence interval for the difference in population means between this year's sales and last year's sales is between -0.608 and 12.608. This suggests that the population mean for this year's sales is likely to be higher than the population mean for last year's sales. This is supported by the fact that the estimate for the difference in population means is 6.000, which is greater than 0.

The logarithmic form of the given exponential equation in the question,     4³ = (5x + 4) is Log₄ (5x + 4) = 3.

The exponential-to-logarithmic format is useful because it makes it easy to compute large numeric expressions. Expressions involving exponential multiplication and division can be converted to addition and subtraction by applying logarithms. Logarithmic exponential form is a common means of converting mathematical expressions in one form to another form.

The exponential form aˣ = N is transformed and written logarithmically as LogₐN = x.

The exponential form of a for the exponent of x equal to N is converted to the logarithm of the number N in base a, equal to x.

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The probability of rolling two even numbers is 1/4.

The probability of rolling one even and one odd number is 1/4.

The probability of flipping tails on a coin and rolling a factor of 6 on a six-sided die is 1/6.

To roll two even numbers, the first roll can be either 2, 4, or 6 (3 even numbers), and the second roll can also be 2, 4, or 6.

The probability of rolling an even number on a six-sided die is 3/6 = 1/2, so the probability of rolling two even numbers is (1/2) x (1/2) = 1/4.

The probability of rolling an odd number on a six-sided die is also 1/2, so the probability of rolling one even and one odd number is (3/6) x (3/6) = 9/36

the probability of rolling one even and one odd number = 1/4.

The probability of flipping tails on a coin is 1/2, and the probability of rolling a factor of 6 (i.e., rolling a 2 or a 3) on a six-sided die is 2/6 = 1/3. Since these two events are independent, we can multiply their probabilities to get the probability of both events happening: (1/2) x (1/3) = 1/6.

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Answer:

Therefore, the person actually needs 148 grams to have a total of 532 grams for Lou's green tea.

Step-by-step explanation:

The statement is incorrect because the person said they have 384 grams, but then they said they still need 200 grams for Lou's green tea, which would mean they need a total of 584 grams.

To find out how much they actually need, we can subtract the amount they have (384 grams) from the amount they need (532 grams) to get:

532 g - 384 g = 148 g

Therefore, the person actually needs 148 grams to have a total of 532 grams for Lou's green tea.

The range of this relation include the following: D. {-2, 3, 5}.

In Mathematics and Geometry, a range can be defined as the set of all real numbers that connects with the elements of a domain. This ultimately implies that, a range simply refers to the set of all possible output numerical values (real numbers), which are shown on the y-coordinate (y-axis) of a graph.

Based on the information provided in this scenario, the domain and range of this relation in interval notation are as follows:

Domain = {3, 6, 8}

Range = {-2, 3, 5}.

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Answer: The empirical rule states that for a normal distribution, approximately 68% of the data falls within one standard deviation of the mean, 95% falls within two standard deviations, and 99.7% falls within three standard deviations.

In this case, we have a value that is more than 5 standard deviations below the mean. Using the formula for z-score, we have:

z = (x - mu) / sigma

z = (3.46 - 8.05) / 1.53

z = -2.98

According to the empirical rule, almost all of the data falls within three standard deviations, so the percentage of American women with shoe sizes less than 3.46 is approximately:

0.15%

Step-by-step explanation:

There is a 0.00567 percent chance that 4 out of the 5 auto damage claims were submitted by individuals under the age of 25.

An expression that has been raised to any finite power can be expanded using the binomial theorem. A binomial theorem is a potent expansionary technique with uses in probability, algebra, and other fields.

A binomial expression is an algebraic expression with two terms that are not the same. For instance, a+b, a3+b3, etc.

Let n = N, x, y, R, then the binomial theorem holds.

(x + y)n = nΣr=0 where, nCr xn - r yr

The likelihood of an event happening is gauged by probability. Several things are impossible to completely predict in advance. Using it, we can only make predictions about how probable an event is to happen, or its chance of happening. The probability might be between 0 and 1, where 0 denotes an impossibility and 1 denotes a certainty. A crucial subject for pupils in class 10, probability explains all the fundamental ideas of the subject. A sample space has an overall probability of 1 for all events.

This is a binomial probability problem, where each automobile damage claim is a Bernoulli trial with a probability of success (a claim made by someone under the age of 25) of p=0.70. We want to find the probability of getting exactly 4 successes out of 5 trials.

The probability of getting exactly k successes out of n trials in a binomial experiment with probability of success p is given by the binomial probability formula:

P(k successes out of n trials) = (n choose k) * [tex]p^k[/tex] * [tex](1-p)^(n-k)[/tex]

where (n choose k) = n! / (k! * (n-k)!) is the number of ways to choose k items out of n items.

In this case, we have n=5 and p=0.70. So, the probability of getting exactly 4 successes out of 5 trials is:

P(4 out of 5 claims made by someone under 25) = (5 choose 4) * [tex]0.70^4[/tex] *[tex](1-0.70)^(5-4)[/tex]

P(4 out of 5 claims made by someone under 25) = 5 * [tex]0.70^4[/tex] *[tex]0.30^1[/tex]

P(4 out of 5 claims made by someone under 25) = 0.00567 (rounded to 5 decimal places)

Therefore, the probability that exactly 4 of the 5 automobile damage claims were made by people under the age of 25 is approximately 0.00567.

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The only equation that is a polynomial identity is (A), (c+d)³ = c³ - d³ + 3cd(c + d).

It should be noted that to determine which equation is a polynomial identity, we need to simplify both sides of each equation and see if they are equal for all values of c and d.

(A) (c+d)³ = c³ + 3c²d + 3cd² + d³

= c³ - d³ + 3cd(c + d)

The left-hand side can be expanded using the binomial formula. The right-hand side is a polynomial of degree 3, so we can see that equation (A) is a polynomial identity.

(B) (c+d) = c + d³ + 3cd(c + d)

= d³ + c³ + 3cd(c + d) + c + d - c³

= d³ + c³ + 3cd(c + d) + c + d

The right-hand side can be simplified, but it is not equal to the left-hand side for all values of c and d. Therefore, equation (B) is not a polynomial identity.

(C) (c+d)³ = c³ + 3c²d + 3cd² + d³

= c³ + d³ + 3cd(c + d) + cd(c - d)

The right-hand side is not equal to the left-hand side for all values of c and d, so equation (C) is not a polynomial identity.

(D) (c+d)³ = c³ + 3c²d + 3cd² + d³

= c³ - d³ + 3cd(c - d) + c³ + d³

= 2c³ + 3cd(c - d)

The right-hand side is not equal to the left-hand side for all values of c and d, so equation (D) is not a polynomial identity.

Therefore, the only equation that is a polynomial identity is (A), (c+d)³ = c³ - d³ + 3cd(c + d).

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21 is  the area, measured in square centimeters, of the green shape pictured below.

What is square?

Square, in geometry, a plane figure with four equal sides and four right (90°) angles. A square is a special kind of rectangle (an equilateral one) and a special kind of parallelogram (an equilateral and equiangular one).

the area of a parallelogram, we can split this shape up into a rectangle and two right triangles.

The rectangle:

The rectangle has dimensions of 4 by 3 centimeters therefore the area would be 4*3=12 cm squared

two right triangles:

The area of a triangle is 1/2 base * height, but since there are two triangles it would just be the base times the height. The base is 3 cm and the height is 3 so 3*3=9

Next, we add the two areas together: 12+9 = 21

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Answer:

1

Step-by-step explanation:

1/3 + 8/12 ⬅️(factor out by common factor, 4)

= 1/3 + 2/3

=1+2/ 3

=3/3 or 1