There are 88 seats in the theater. The seating in the theater is split in to 4 identical section has 14 red seats and some blue seats.

Answer:

A. 6

Step-by-step explanation:

To solve the equation 6x^2-2x+36=5x^2+10x, we can follow these steps:

Move all the terms to one side of the equation by subtracting 5x^2 and 10x from both sides:

6x^2 - 2x + 36 - 5x^2 - 10x = 0

Simplifying the left side:

x^2 - 12x + 36 = 0

Factor the quadratic expression on the left side of the equation:

(x - 6)(x - 6) = 0

Apply the zero product property, which states that if the product of two factors is equal to zero, then at least one of the factors must be zero:

x - 6 = 0

Solve for x:

x = 6

The solution to the equation 6x^2-2x+36=5x^2+10x is x = 6.

Answer:

The height of the building is approximately 55 meters.

Step-by-step explanation:

Let's call the height of the building "h" and the distance from point P to the foot of the building "d".

According to the problem, we have:

d = 50m

T = 23.6m

Using the Pythagorean theorem, we know that:

h^2 = T^2 + d^2

Substituting the values we have:

h^2 = (23.6m)^2 + (50m)^2

h^2 = 556.96m^2 + 2500m^2

h^2 = 3056.96m^2

Taking the square root of both sides, we get:

h = sqrt(3056.96m^2)

h = 55.28m

Rounding to the nearest meter, we get:

h ≈ 55m

Therefore, the height of the building is approximately 55 meters.

As a result, 25% of chlorine is present in solution B.

It's 100 × 20 / 100 = 20% ! In this circumstance, percentages are useful. When describing a change from one % to another, we use percentage points. The difference between 10% and 12% is two percent respectively (or 20 percent).

Let x represent the chlorine content of solution B.

The entire quantity of chlorine inside the combination can be used to create the following equation as a starting point:

3 cups of solution A × 10% chlorine + 6 cups of solution B × x% chlorine = (3+6) cups of new solution × 20% chlorine

When we simplify this equation, we obtain:

0.3 + 6x = 1.8

By taking 0.3 away both from sides, we get at:

6x = 1.5

When we multiply both parts by 6, we get:

x = 0.25

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The percentage that is less than the average number of shoppers in the original store at any time is 60%.

Little's law is a fundamental principle in queueing theory that relates the average number of customers in a stable system to the average time that a customer spends in the system.

The law states that the average number of customers N in the system is equal to the average rate of customer arrivals r multiplied by the average time W that a customer spends in the system:

Here we have

For the new store, the owner estimates that, during business hours, an average of 90 shoppers per hour enter the store and each of them stays an average of 12 minutes.

=> Number of shoppers per minute = 1.5    

=> Rate of shoppers per minute = 1.5

The manager estimates that each shopper stays in the store for an average of 12 minutes.

Hence, by Little’s law, the number of shoppers N = r × t

=> Number of shoppers = (1.5) × 12 = 18  

Let the estimated average number of shoppers in the original store at any time be 45.    

So, the number of shoppers is (45 - 18) less than the original i.e 27

Percentage  [ 27/45 ] × 100 = 60%  

Therefore,

The percentage that is less than the average number of shoppers in the original store at any time is 60%.

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The real zeros of this function are 0 (multiplicity 4),-8 (multiplicity 2), and 2 (multiplicity 4).

The real zeros of the function occur when any of the factors is equal to zero:

x^(4) = 0, (x-2)^(4) = 0, or (x+8)^(2) = 0

To find the real zeros of the given function, we need to set the function equal to zero and solve for x:

f(x) = x^(4)(x-2)^(4)(x+8)^(2) = 0

The process to find these zeros is as follows:

1. Set f(x) = 0 and solve for x

2. Factor the polynomial and solve each factor:

f(x) = 0 => x^(4)(x-2)^(4)(x+8)^(2) = 0
           => x4 = 0
           => x = 0 (multiplicity 4)

           => (x-2)4 = 0
           => x = 2 (multiplicity 4)

           => (x+8)2 = 0
           => x = -8 (multiplicity 2)

The multiplicity of a zero is the number of times it appears as a factor in the function. In this case, the zero 0 has a multiplicity of 4, the zero 2 has a multiplicity of 4, and the zero -8 has a multiplicity of 2.

Therefore, the real zeros and their multiplicities are:
0 with a multiplicity of 4
2 with a multiplicity of 4

-8 with a multiplicity of 2

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The sample proportions of male births that are at least as extreme as the sample proportion of 472/960 are 0.0026 for both the lower and upper tails.

A proportion is an equation in which two ratios are set equal to each other.

Part 1:

a. The values of p and q can be identified as follows:

p = proportion of male births in the population = 0.512

q = proportion of female births in the population = 1 - p = 1 - 0.512 = 0.488

So, p = 0.512 and q = 0.488.

Part 2:

b. The sample proportion of male births is:

P = 472/960 = 0.4917

To find the sample proportions of male births that are at least as extreme as this value, we need to calculate the z-scores corresponding to the upper and lower tails of the distribution under the null hypothesis (i.e., p = 0.512). The formula for the z-score is:

z = (P - p) / √(p*q/n)

where n is the sample size.

For the lower tail, we have:

z = (0.4917 - 0.512) / √(0.512*0.488/960) = -2.79

For the upper tail, we have:

z = (0.512 - 0.4917) / √(0.512*0.488/960) = 2.79

Using a standard normal distribution table or calculator, we can find the probabilities associated with these z-scores:

P(z < -2.79) = 0.0026

P(z > 2.79) = 0.0026

Hence, the sample proportions of male births that are at least as extreme as the sample proportion of 472/960 are 0.0026 for both the lower and upper tails.

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Complete Question:

A random sample of 960 births in New York State included 472 ​boys and that sample is to be used for a test of the common belief that the proportion of male births in the population is equal to

0.512.

Complete parts​ (a) through​ (c).

Question content area bottom

Part 1

a. In testing the common belief that the proportion of male babies is equal to 0.512​, identify the values of p and p. p=enter your response here p=enter your response here

Part 2

b. For random samples of size 960​,

what sample proportions of male births are at least as extreme as the sample proportion of 472960​?

The answer is 3y.

Repeat the question in your answer. "We need to subtract the polynomials (7y+5)-(4y+5)."

Subtract the terms with the same variable and the same degree. In this case, you need to subtract 7y and 4y, and 5 and 5.

Write the subtraction in the form of an equation.

"7y - 4y = 3y" and "5 - 5 = 0"

Write the final answer. "The result of subtracting the polynomials is 3y + 0, or simply 3y."

So the final answer is 3y.

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The statement  that best describes the process to find the product of rational expressions is: D. Multiply the numerators and then multiply the denominators.

The process to find the product of rational expressions involves multiplying the two expressions together. This can be done by multiplying the numerators of the two expressions and then multiplying the denominators of the two expressions.

For example, if we want to find the product of the rational expressions (2x + 3)/(x - 1) and (x + 2)/(3x), we can do the following:

(2x + 3)/(x - 1) * (x + 2)/(3x) = (2x + 3)(x + 2) / (x - 1)(3x)

To find the product, we multiplied the numerators (2x + 3) and (x + 2) to get (2x + 3)(x + 2), and we multiplied the denominators (x - 1) and (3x) to get (x - 1)(3x).

Therefore, the correct option is: Multiply the numerators and then multiply the denominators.

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19). The probability that a student attained a B or a C on the quiz is 0.42, or 42%. 20). The events for selecting the balls are dependent. This is because the probability of selecting the second ball is affected by the selection of the first ball.

The probability that a student attained a B or a C on the quiz can be found by adding the probabilities of the two events together. First, find the probability of attaining a B by dividing the number of students who attained a B by the total number of students. Do the same for the probability of attaining a C. Then add the two probabilities together to find the total probability.

Probability of attaining a B = (Number of students who attained a B) / (Total number of students) = 17 / 100 = 0.17
Probability of attaining a C = (Number of students who attained a C) / (Total number of students) = 25 / 100 = 0.25
Total probability = 0.17 + 0.25 = 0.42
Therefore, the probability that a student attained a B or a C on the quiz is 0.42, or 42%.

After the first ball is selected, there is one less ball in the bin, which changes the probability of selecting the second ball. Therefore, the events are dependent.

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1 at each vertex

An exterior angle of a polygon is an angle formed by a side of the polygon and the extension of an adjacent side. In the case of a triangle, each exterior angle is formed by one of the triangle's sides and the extension of an adjacent side.

When an exterior angle is formed at a vertex of a polygon, the measure of the exterior angle is equal to the sum of the measures of the two interior angles adjacent to it. In the case of a triangle, the sum of the measures of the two interior angles adjacent to the exterior angle is always 180 degrees (which is the sum of the measures of all three interior angles of a triangle).

Since each exterior angle of a triangle is formed by two interior angles, and the sum of the measures of those interior angles is always 180 degrees, there can only be one exterior angle at each vertex of a triangle. Therefore, a triangle has one exterior angle at each vertex.

The side of figure WXYZ corresponds with QR is XY

In mathematics, mirror images are often used in the study of symmetry and geometry. A mirror image, also known as a reflection, is a transformation that flips an object over a line called the mirror line.

An image appears to be reversed from left to right. Therefore, the left side of the object appears to be on the right side of the image, and the right side of the object appears to be on the left side

Here we have

Parallelogram PQRS and WXYZ which are two mirror images

Here the corresponding side of the two figures are

=> PQ and WX

=> QR and XY

=> RS and YZ

=> PS and WZ

Therefore,

The side of figure WXYZ corresponds with QR is XY

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Local Minima is -1.

A. f has a local minimum at x = -1; the local minimum is -1.

To find the local minima of a function, we must identify any points at which the slope of the graph is equal to zero. In this graph, the slope of the graph is equal to zero at x = -1, indicating that f has a local minimum at x = -1. The value of the function at this point is -1, so the local minimum is -1.

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The standard form of the equation of a quadratic function with roots of 4 and −1 that passes through (1, −9) is [tex]y = 1.5x^{2} - 4.5x - 6[/tex]

What is the quadratic function?

A quadratic function is a type of function that can be written in the form:

[tex]f(x) = ax^2 + bx + c[/tex]

where a, b, and c are constants, and x is the variable. This function is a second-degree polynomial function, which means that the highest power of the variable x is 2.

Quadratic functions can be graphed as a U-shaped curve called a parabola. The sign of the coefficient a determines whether the parabola opens up or down. If a > 0, the parabola opens up, and if a < 0, the parabola opens down. The vertex of the parabola is the minimum or maximum point of the function, depending on whether the parabola opens up or down.

Quadratic functions are used in many areas of mathematics, science, and engineering to model various phenomena such as projectile motion, population growth, and optimization problems.

To write the standard form of the equation of a quadratic function, we need to use the roots of the function and another point on the curve. The standard form of the quadratic function is:

y = a(x - r1)(x - r2)

where r1 and r2 are the roots of the quadratic function, and a is a constant.

Given that the roots of the quadratic function are 4 and -1, we can write:

y = a(x - 4)(x + 1)

To find the value of a, we can use the point (1, -9) that the function passes through:

-9 = a(1 - 4)(1 + 1)

-9 = -6a

a = 3/2

Substituting this value of a in the equation, we get:

[tex]y = 1.5(x - 4)(x + 1)[/tex]

Expanding this equation, we get:

[tex]y = 1.5x^{2} - 4.5x - 6[/tex]

Therefore, the standard form of the equation of the quadratic function with roots of 4 and −1 that passes through (1, −9) is [tex]y = 1.5x^{2} - 4.5x - 6[/tex]

So, the correct answer is: [tex]y = 1.5x^{2} - 4.5x - 6[/tex]

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Answer:

A

Step-by-step explanation:

sorry if it is wrong

The solution set of the given inequality is (164/18, ∞).

To solve the inequality symbolically, we first need to isolate the variable x on one side of the inequality. We can do this by multiplying both sides by 8 and then subtracting 4 from both sides. Finally, we can divide both sides by 6 to solve for x. Here are the steps:

(6x + 4)/8 > 22/3

6x + 4 > (22/3)(8)

6x + 4 > 176/3

6x > (176/3) - 4

6x > 164/3

x > (164/3)(1/6)

x > 164/18

Now we can express the solution set x > 164/18 in interval notation: (164/18, ∞)

So the solution set is all values of x greater than 164/18.

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Answer:

4 : 2 : 5

Step-by-step explanation:

To find the ratio of silver to brown to pink counters in simplest form, we need to divide the number of each type of counter by their greatest common factor.

The greatest common factor of 16, 8, and 20 is 4.

So, we divide each of the numbers by 4:

Silver counters: 16 ÷ 4 = 4

Brown counters: 8 ÷ 4 = 2

Pink counters: 20 ÷ 4 = 5

Therefore, the ratio of silver to brown to pink counters in simplest form is:

4 : 2 : 5

or

2 : 1 : 2.5 (if we prefer to express the ratio in decimal form)

Substituting the values into the formula gives us Angle XZY = 180° - angle XYZ - angle YXZ = 180° - 95° - 50° = 35°. Thus, the measure of angle XZY in triangle XYZ is 35 degrees

We use the knowledge that the total of the angles in any triangle is always 180 degrees to determine the size of angle XZY in triangle XYZ. Angle YXZ is known to measure 50 degrees, while angle XYZ is known to measure 95 degrees. In order to determine the measure of angle XZY, we can subtraction the measurements of these two angles from 180 degrees. We obtain Angle XZY = 180° - angle XYZ - angle YXZ = 180° - 95° - 50° = 35° by substituting the values into the formula. As a result, the angle XZY in triangle XYZ has a measure of 35 degrees.

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Answer:

The stock market goes up 70 1/2 points at the beginning of the day, so we can represent the starting value as:

Starting value = 70 1/2

The stock market stays at this level for most of the day, so the value remains the same until the end of the day.

At the end of the day, the stock market goes down 120 3/4 points from the high at the beginning of the day. To calculate the ending value, we need to subtract this decrease from the starting value:

Ending value = Starting value - Decrease

Ending value = 70 1/2 - 120 3/4

To subtract the two values, we need to convert them to a common fraction with a common denominator of 4:

70 1/2 = 141/2

120 3/4 = 483/4

Ending value = 141/2 - 483/4

Ending value = 282/4 - 483/4

Ending value = -201/4

Therefore, the total change in the stock market from the beginning of the day to the end of the day is:

Ending value - Starting value = (-201/4) - (141/2) = -201/4 - 282/4 = -483/4

The total change in the stock market from the beginning of the day to the end of the day is a decrease of 483/4 points.

The value of  ∂u∂f​​(u,v)=(−2,−2) = -20 and ∂v∂f​​(u,v)=(−2,−2) = -45.

To evaluate the partial derivatives ∂u∂f​​(u,v)=(−2,−2) and ∂v∂f​​(u,v)=(−2,−2), we can use the Chain Rule. We first take the partial derivatives of f(x,y,z) with respect to x, y, and z, and then substitute in x = u2 + v, y = u + v2, and z = 4uv.

First, ∂f∂x​=3x2, ∂f∂y​=yz and ∂f∂z​=y2.

Substituting in x = u2 + v, y = u + v2, and z = 4uv gives us: ∂f∂x​=3(u2 + v)2, ∂f∂y​=(u + v2)(4uv) and ∂f∂z​=(u + v2)2.

Next, we use the Chain Rule to find ∂u∂f​​(u,v)=(−2,−2) and ∂v∂f​​(u,v)=(−2,−2):

∂u∂f​​(u,v)=(−2,−2)= ∂f∂x​•∂x∂u​​ + ∂f∂y​•∂y∂u​​ + ∂f∂z​•∂z∂u​​ = 3(u2 + v)2•(2u) + (u + v2)(4uv)•(1) + (u + v2)2•(4v) = 6u2 + 8uv + 4uv + 4v2 = 10uv + 6u2 + 4v2

∂v∂f​​(u,v)=(−2,−2)= ∂f∂x​•∂x∂v​​ + ∂f∂y​•∂y∂v​​ + ∂f∂z​•∂z∂v​​ = 3(u2 + v)2•(1) + (u + v2)(4uv)•(2v) + (u + v2)2•(4u) = 3 + 8uv + 8u2 = 8u2 + 8uv + 3

When (u,v)=(−2,−2), we have ∂u∂f​​(u,v)=(−2,−2) = 10(-2)(-2) + 6(-2)2 + 4(-2)2 = 20 - 24 - 16 = -20 and ∂v∂f​​(u,v)=(−2,−2) = 8(-2)2 + 8(-2)(-2) + 3 = -32 - 16 + 3 = -45.

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Answer: [tex]\sqrt{3}[/tex]

Step-by-step explanation:

take,

x^2 = Y

6Y^2 + 8Y = 26Y

6Y^2 + 8Y - 26Y =  0

6Y^2 - 18Y = 0

6Y (Y - 3) = 0

(Y - 3) = 0

Y  = 3

x^2 = y

x^2 = 3

x = [tex]\sqrt{3}[/tex]

The domain of \(f(x)=\tan ^{-1} x\) is \(\mathbb{R}\) and the range is \((-\frac{\pi}{2}, \frac{\pi}{2})\).

Explanation:

The domain of a function is the set of all possible input values for which the function is defined. The inverse tangent function, \(f(x)=\tan ^{-1} x\), is defined for all real numbers, so the domain is \(\mathbb{R}\).

The range of a function is the set of all possible output values for which the function is defined. The inverse tangent function, \(f(x)=\tan ^{-1} x\), has a range of \((-\frac{\pi}{2}, \frac{\pi}{2})\). This is because the tangent function has a period of \(\pi\), and the inverse tangent function is the inverse of the tangent function restricted to the interval \((-\frac{\pi}{2}, \frac{\pi}{2})\).

Therefore, the domain of \(f(x)=\tan ^{-1} x\) is \(\mathbb{R}\) and the range is \((-\frac{\pi}{2}, \frac{\pi}{2})\).

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Answer:

9 [tex]cm^{2}[/tex]

Step-by-step explanation:

All the sides have the same length, so if the perimeter.  The distance around the figure is 24, then each side is 6 (24 divided by 4 is 6)

The area of a triangle is

a = 1/2 bh

a = 1/2(6)(3)

a = 1/2(18)

a = 9

Helping in the name of Jesus.

Anne prediction on the amount of rain that will pour down is 60 inches

From the question, we have the following parameters that can be used in our computation:

Using the above as a guide, we have the following:

The amount of rain this year can be calculated as follows

Percentage = 20/100 = 0.2

So, we have

Proportion = 0.2 + 1  =  1.2

This gives

Amount = 1.2 × 50 = 60

Hence the amount of rain this year is 60 inches

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The given set of vectors is: \( \left\{\left[\begin{array}{c}1 \\ 0 \\ -2\end{array}\right],\left[\begin{array}{c}-1 \\ 1 \\ 4\end{array}\right],\left[\begin{array}{c}1 \\ 2 \\ -2\end{array}\right]\right\} \)

To determine if the given set of vectors is linearly independent or linearly dependent, we can use the determinant method. We will form a matrix using the given vectors as columns and then find the determinant of the matrix. If the determinant is zero, then the vectors are linearly dependent. If the determinant is not zero, then the vectors are linearly independent.

The matrix formed using the given vectors as columns is:
\[ \left[\begin{array}{ccc}1 & -1 & 1 \\ 0 & 1 & 2 \\ -2 & 4 & -2\end{array}\right] \]
The determinant of the matrix is:
\[ \begin{vmatrix}1 & -1 & 1 \\ 0 & 1 & 2 \\ -2 & 4 & -2\end{vmatrix} = (1)(1)(-2) + (-1)(2)(-2) + (1)(0)(4) - (1)(2)(4) - (-1)(0)(-2) - (1)(1)(-2) = -2 + 4 + 0 - 8 + 0 + 2 = -4 \]

Since the determinant is not zero, the given set of vectors is linearly independent.

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For subspace (a.1) basis for Col(A) is the first four columns of A and dimension is 4. For (a.2) basis for Row(A) is the first four rows of A and dimension is 4. A basis for (a.3) Nul(A) is the set of vectors obtained by setting one free variable to 1 and the others to 0 and dimension is 7.

A basis for a subspace is a set of vectors that are linearly independent and span the subspace. The dimension of a subspace is the number of vectors in a basis for that subspace.

(a.1) Col(A) is the subspace of R^4 spanned by the columns of A. To find a basis for Col(A), we can reduce A to its reduced row echelon form (RREF) and find the columns of A that correspond to the pivot columns in the RREF. The RREF of A is:

1 0 0 0 0 0 0 0 0 0 0
0 1 0 0 0 0 0 0 0 0 0
0 0 1 0 0 0 0 0 0 0 0
0 0 0 1 0 0 0 0 0 0 0

The pivot columns are the first four columns, so a basis for Col(A) is the first four columns of A:

{[1, 1, 1, 1], [1, 1, -1, -1], [1, -1, 1, -1], [1, -1, -1, 1]}

The dimension of Col(A) is the number of vectors in the basis, which is 4.

(a.2) Row(A) is the subspace of R^11 spanned by the rows of A. To find a basis for Row(A), we can reduce A to its RREF and find the nonzero rows. The RREF of A is the same as above, so a basis for Row(A) is the first four rows of A:

{[1, 1, 1, 1, 1, 1, 1, 1, 1, 1, -1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2], [0, 0, 0, 0, 0, 0, 0, 1, -1, 0, 0], [0, 0, 0, 0, 0, 0, 1, -3, 3, 0, 0]}

The dimension of Row(A) is the number of vectors in the basis, which is 4.

(a.3) Nul(A) is the subspace of R^11 consisting of all vectors x such that Ax = 0. To find a basis for Nul(A), we can reduce A to its RREF and find the solutions to the homogeneous equation Ax = 0. The RREF of A is the same as above, and the general solution to Ax = 0 is:

x1 = 0
x2 = 0
x3 = 0
x4 = 0
x5 = free
x6 = free
x7 = free
x8 = free
x9 = free
x10 = free
x11 = free

A basis for Nul(A) is the set of vectors obtained by setting one free variable to 1 and the others to 0:

{[0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1]}

The dimension of Nul(A) is the number of vectors in the basis, which is 7.

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Answer:

1+1= 2

Step-by-step explanation:

im a genuis

47. The identity that expresses cot(x) in terms of csc(x) is: cot(x) = 1/sin(x) = csc(x)/sin(x) * sin(x)/sin(x) = csc(x)cos(x)

48. The identity that expresses sec(x) in terms of tan(x) is: sec(x) = 1/cos(x) = 1/cos(x) * sin(x)/sin(x) = sin(x)/(cos(x)sin(x)) = sin(x)/sin(x)cos(x) = 1/cos(x) = sec(x)

49. The identity that expresses sin(x) in terms of cot(x) is: sin(x) = 1/csc(x) = 1/csc(x) * cos(x)/cos(x) = cos(x)/(csc(x)cos(x)) = cos(x)/cos(x)csc(x) = 1/csc(x) = sin(x)

50. The identity that expresses cos(x) in terms of tan(x) is: cos(x) = 1/sec(x) = 1/sec(x) * cos(x)/cos(x) = cos(x)/(sec(x)cos(x)) = cos(x)/cos(x)sec(x) = 1/sec(x) = cos(x)

51. The identity that expresses tan(x) in terms of csc(x) is: tan(x) = sin(x)/cos(x) = sin(x)/cos(x) * 1/csc(x) = sin(x)csc(x)/cos(x) = 1/cos(x) = sec(x)

52. The identity that expresses cot(x) in terms of sec(x) is: cot(x) = 1/tan(x) = 1/(sin(x)/cos(x)) = cos(x)/sin(x) = cos(x)/sin(x) * 1/sec(x) = cos(x)sec(x)/sin(x) = 1/sin(x) = csc(x)

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The values of y that make the inequality 5y<41 true are B. 7 and D. 6.5.

To find the values of y that make the inequality true, we can first isolate y by dividing both sides of the inequality by 5:

5y<41

y<41/5

y<8.2

This means that any value of y less than 8.2 will make the inequality true.

Looking at the given options, we can see that B. 7 and D. 6.5 are both less than 8.2, so they are the correct answers.

A. 8 and C. 8.5 are both greater than or equal to 8.2, so they do not make the inequality true. E. 9 is also greater than 8.2, so it does not make the inequality true.

Therefore, the correct answers are B. 7 and D. 6.5.

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The answers are a.(f∘g)(x) = x^2+2x+1 and b. (g∘f)(x) = x^2+1.

Let f(x)=x^2 and g(x)=x+1. We are asked to find a.(f∘g)(x) and b. (g∘f)(x).

a. (f∘g)(x) = f(g(x)) = f(x+1) = (x+1)^2 = x^2+2x+1

b. (g∘f)(x) = g(f(x)) = g(x^2) = x^2+1

Therefore, the answers are a.(f∘g)(x) = x^2+2x+1 and b. (g∘f)(x) = x^2+1.

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The area of John's actual bedroom is 120 square feet.

What is Scale factor?

Scale factor is a mathematical concept that is used to describe the ratio of corresponding dimensions of two similar figures. In geometry, two figures are said to be similar if they have the same shape but possibly different sizes. The scale factor is the ratio of the length of a side (or any corresponding dimension) of one figure to the length of the corresponding side (or dimension) of the other figure.

If John used a scale of 3 inches = 2 feet, then we can convert the dimensions of the drawing to the actual dimensions of his bedroom using the scale factor:

1 inch on the drawing corresponds to 2/3 feet in reality.

So, the actual width of the bedroom is:

15 inches × (2/3 feet/inch) = 10 feet

And the actual length of the bedroom is:

18 inches × (2/3 feet/inch) = 12 feet

The area of the bedroom is the product of its actual length and width:

Area = length × width = 12 feet × 10 feet = 120 square feet

Therefore, the area of John's actual bedroom is 120 square feet.

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