5. Tony measured a TV that is approximately 14 inches tall by 24 inches wide. Since the size of TVs are named by the measure of the diagonal, what is the BEST diagonal measure, in inches, for this TV?

A. 772 inches
B. 28 inches
C. 76 inches
D. 9 inches

Answer:

6. From speed vs. time graphs, we know that in order to find the distance, we need to look at the area covered. Given the rectangular shape below, we know that it would be A=lw so A=60mph(2.5h).

7. A=60mph(2.5h)= 150 miles.

c = 1/2, and the pdf is fx(x) = { 1/2 [tex](1-x^{2} )^{-1}[/tex], −1 ≤ x ≤ 1or 0 otherwise and cdf of X is: FX(x) = { 0, x ≤ -1 or -1/2 [tex]tan^{-1}(x/\sqrt{1-x^{2} )}[/tex] + 1/2, -1 < x < 1 or 1 ,x ≥ 1

(a) To find c, we need to integrate the pdf over its support and set the result equal to 1 since the pdf must integrate to 1 over its entire support. Therefore, we have:

1 = ∫c [tex](1-x^{2} )^{-1}[/tex] dx from -1 to 1

Using the substitution u = 1 - [tex]x^{2}[/tex], we have:

1 = c∫ [tex]u^{-1/2}[/tex] dx from 0 to 1

Solving the integral, we get:

1 = 2c

Therefore, c = 1/2, and the pdf is:

fx(x) = {

1/2 [tex](1-x^{2} )^{-1}[/tex] , −1 ≤ x ≤ 1

0 otherwise

To sketch the pdf, we can notice that it is symmetric about x = 0 and that it approaches infinity as x approaches ±1. Therefore, it will have a peak at x = 0 and decrease as we move away from x = 0 in either direction.

(b) To find the cdf of X, we can integrate the pdf from negative infinity to x for each value of x in the support of the pdf. Therefore, we have:

FX(x) = ∫fX(t) dt from -∞ to x

For x ≤ -1, FX(x) = 0, since the pdf is zero for x ≤ -1. For -1 < x < 1, we have:

FX(x) = ∫1/2 [tex](1-t^{2} )^{-1}[/tex] dt from -1 to x

Using the substitution u = [tex]t^{2}[/tex] - 1, we have:

FX(x) = -1/2 ∫[tex](x^{2} -1)^{-1/2}[/tex] du

Solving the integral, we get:

FX(x) =  -1/2 [tex]tan^{-1}(x/\sqrt{1-x^{2} )}[/tex]  + 1/2

For x ≥ 1, we have FX(x) = 1, since the pdf is zero for x ≥ 1. Therefore, the cdf of X is:

FX(x) = {

0 x ≤ -1

-1/2 [tex]tan^{-1}(x/\sqrt{1-x^{2} )}[/tex]  + 1/2 -1 < x < 1

1 x ≥ 1

To sketch the cdf, we can notice that it starts at 0 and increases gradually from x = -1 to x = 1, where it jumps to 1. The cdf is also symmetric about x = 0, similar to the pdf.

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The area under the curve of the function f(x) between the points x = 1 and x = 5 is equal to 6 units.

Given that f is a continuous function and ∫1 5 f(x)dx = 6, we know that the area under the curve of f between 1 and 5 is equal to 6.

We don't have any information about g, so we can't say much about it. However, we can use the fact that f is continuous to make some deductions.

Since f is continuous, we know that it must be integrable on [1, 5]. This means that we can define another function F(x) as the definite integral of f(x) from 1 to x:

F(x) = ∫1 x f(t) dt

We can then apply the fundamental theorem of calculus to F(x) to get:

F'(x) = f(x)

In other words, F(x) is an antiderivative of f(x), so its derivative is f(x).

We can also use the fact that f is continuous to say that it must have an average value on [1, 5]. This average value is given by:

avg(f) = (1/4) ∫1 5 f(x) dx

Using the fact that ∫1 5 f(x)dx = 6, we can simplify this to:

avg(f) = (1/4) * 6 = 1.5

This means that on average, the value of f(x) between 1 and 5 is 1.5.

However, since we don't know anything else about f, we can't say much more than that.


Based on the given information, f and g are continuous functions and the definite integral of f(x) from 1 to 5 is equal to 6. In mathematical notation, it can be expressed as:

∫(from 1 to 5) f(x)dx = 6

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The expression 7 - 3(2x + 5) is equivalent to -6x - 8.

Given the expression in the question:

7 - 3(2x + 5)

We can simplify this expression using the distributive property of multiplication over addition or subtraction.

According to this property, when a number is multiplied by a sum or difference, we can distribute the multiplication over each term within the parentheses.

So, applying the distributive property, we get:

7 - 3(2x + 5)

7 - 3 × 2x -3 × 5

= 7 - 6x - 15

Simplifying further, we can combine like terms:

= -6x - 8

Therefore, the expression to -6x - 8, which is option C.

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Answer:

x = 9

Step-by-step explanation:

since the triangles are similar then the ratios of corresponding sides are in proportion, that is

[tex]\frac{BC}{DE}[/tex] = [tex]\frac{AB}{AD}[/tex] ( substitute values )

[tex]\frac{2x+3}{15}[/tex] = [tex]\frac{20+8}{20}[/tex] = [tex]\frac{28}{20}[/tex] ( cross- multiply )

20(2x + 3) = 15 × 28 = 420 ( divide both sides by 20 )

2x + 3 = 21 ( subtract 3 from both sides )

2x = 18 ( divide both sides by 2 )

x = 9

Answer:

x = 9

Step-by-step explanation:

In similar triangles, the corresponding sides are in same proportion.

 AB = AD + DB

       = 20 + 8

       =28

        ΔABC ~ ΔADE,

     [tex]\sf \dfrac{BC}{DE}=\dfrac{AB}{AD}\\\\\\\dfrac{2x + 3}{15}=\dfrac{28}{20}\\\\\ 2x + 3=\dfrac{28}{20}*15[/tex]

     [tex]2x + 3 = 7[/tex] *3

    2x + 3 = 21

Subtract 3 from both sides,

          2x = 21 - 3

          2x = 18

Divide both sides by 2,

            x = 18 ÷ 2

            x = 9

The proportion of samples that will have means between 115 and 120 is 0.4772 or 47.72%.

To solve this problem, we need to use the Central Limit Theorem, which states that the sample means of a large sample size (n>30) from any population with a finite mean and variance will be approximately normally distributed with a mean equal to the population mean (μ) and a standard deviation equal to the population standard deviation (σ/sqrt(n)).

Here, the population mean (μ) = 120, and the population standard deviation (σ) = 10. We are given that the sample size (n) = 16, and we need to find the proportion of samples that will have means between 115 and 120.

First, we need to standardize the sample means using the z-score formula:

z = (x - μ) / (σ / sqrt(n))

where x is the sample mean.

For the lower limit of 115:

z1 = (115 - 120) / (10 / sqrt(16)) = -2

For the upper limit of 120:

z2 = (120 - 120) / (10 / sqrt(16)) = 0

We need to find the area under the standard normal distribution curve between z1 = -2 and z2 = 0. We can use a standard normal distribution table or a calculator to find this area.

Using a standard normal distribution table, we find that the area between z1 = -2 and z2 = 0 is 0.4772.

Therefore, the proportion of samples that will have means between 115 and 120 is 0.4772 or 47.72%.

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The proportion of samples that will have means between 115 and 120 is 0.4772 or 47.72%.

To solve this problem, we need to use the Central Limit Theorem, which states that the sample means of a large sample size (n>30) from any population with a finite mean and variance will be approximately normally distributed with a mean equal to the population mean (μ) and a standard deviation equal to the population standard deviation (σ/sqrt(n)).

Here, the population mean (μ) = 120, and the population standard deviation (σ) = 10. We are given that the sample size (n) = 16, and we need to find the proportion of samples that will have means between 115 and 120.

First, we need to standardize the sample means using the z-score formula:

z = (x - μ) / (σ / sqrt(n))

where x is the sample mean.

For the lower limit of 115:

z1 = (115 - 120) / (10 / sqrt(16)) = -2

For the upper limit of 120:

z2 = (120 - 120) / (10 / sqrt(16)) = 0

We need to find the area under the standard normal distribution curve between z1 = -2 and z2 = 0. We can use a standard normal distribution table or a calculator to find this area.

Using a standard normal distribution table, we find that the area between z1 = -2 and z2 = 0 is 0.4772.

Therefore, the proportion of samples that will have means between 115 and 120 is 0.4772 or 47.72%.

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In the expression, the values are:

a) k = 2.4

b) c = -141.0625

We have the expression:

a√(b² − c) = 5k

a)For the value of ‘k’ when a = 3, b = 6 and c = 20, we have:

a√(b² − c) = 5k

3√(6² − 20) = 5k             (solve for k)

3√(36 − 20) = 5k

3√(16) = 5k  

3*4 = 5k  

12 = 5k

k = 12/5

k = 2.4

b) For the value of ‘c’ when a = 4, b = 7 and k = 11, we have:

a√(b² − c) = 5k

4√(7² − c) = 5*11          (solve for c)

4√(49 − c) = 55

√(49 − c) = 55/4

√(49 − c) = 13.75

49 − c = 13.75²

49 - c = 189.0625

c = 49 - 189.0625

c = -141.0625

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(a). 40,320 different arrangements can be made arranging the letters of FABULOUS.

(b). 5,760 different arrangements have the A appearing anywhere before the S (such as in FABULOUS).

(c). 1,440 different arrangements have the first U appearing anywhere before the S (such as in FABU- LOUS).

(d). 120 different arrangements have all four vowels appear consecutively (such as FAUOUBLS).

(a). The word FABULOUS has 8 letters, so there are 8! = 40,320 different arrangements of its letters.

(b). To count the number of arrangements where A appears before S, we can fix A in the first position and S in the last position. Then, we have 6 remaining letters to arrange in the 6 remaining positions. This gives us 6! = 720 possible arrangements where A appears before S.

However, we can also fix A in the second position and S in the last position, and we can fix A in the third position and S in the last position, and so on. Therefore, the total number of arrangements where A appears anywhere before S is 720 * 8 = 5,760.

(c). To count the number of arrangements where the first U appears before S, we can fix U in the first position and S in the last position, and then we have 6 remaining letters to arrange in the 6 remaining positions. This gives us 6! = 720 possible arrangements where the first U appears before S.

However, there are two U's in the word FABULOUS, so we can also fix the second U in the first position and S in the last position, and then we have 6 remaining letters to arrange in the 6 remaining positions. This also gives us 6! = 720 possible arrangements where the first U appears before S. Therefore, the total number of arrangements where the first U appears anywhere before S is 720 * 2 = 1,440.

(d). To count the number of arrangements where all four vowels appear consecutively, we can group the vowels together as one unit, so we have F, B, L, S, and the group AUOU. The group AUOU has 4 letters, so there are 4! = 24 different arrangements of these letters.

However, the group AUOU can appear in any of the 5 positions between F and B, between B and L, between L and S, after S, or before F. Therefore, the total number of arrangements where all four vowels appear consecutively is 24 * 5 = 120.

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The required answer is Probability = 100 / 100 = 1

If one test subject is randomly selected, the probability that they tested negative or used marijuana can be found using the formula:

P(Negative or Marijuana) = P(Negative) + P(Marijuana) - P(Negative and Marijuana)

Assuming that the test is accurate, the probability of testing negative for marijuana use is typically high for non-users. Let's assume that this probability is 0.95.

On the other hand, the probability of testing positive for marijuana use is typically high for users. Let's assume that this probability is 0.75.

If we assume that the proportion of marijuana users in the sample is 0.2, we can use this to calculate the probability of a subject being a marijuana user and testing negative (0.05 x 0.2 = 0.01).

Using these probabilities, we can calculate the probability that the subject tested negative or used marijuana:

P(Negative or Marijuana) = 0.95 + 0.2 - 0.01 = 1.14

probabilities cannot be greater than 1, so we must adjust this probability by subtracting the probability of both events occurring (i.e., a marijuana user testing negative):

P(Negative or Marijuana) = 0.95 + 0.2 - 0.01 - 0.15 = 1.09
Therefore, the probability that the subject tested negative or used marijuana is 1.09 (or 109%). This is an incorrect probability, as probabilities cannot be greater than 1. It is likely that there is an error in the assumptions or calculations made, and further analysis is needed.

To find the probability that the subject tested negative or used marijuana, you will need to follow these steps:

This mathematical definition of probability can extend to infinite sample spaces, and even uncountable sample spaces, using the concept of a measure.
Step 1: Determine the total number of test subjects.
In this case, the total number of test subjects is not given, so we'll assume it's 100 for simplicity.

Step 2: Determine the number of subjects who tested negative and the number of subjects who used marijuana.
Let's assume 'x' test subjects tested negative and 'y' test subjects used marijuana. Since these are the only two options, x + y = 100.

Step 3: Calculate the probability that the subject tested negative or used marijuana.
The probability that the subject tested negative or used marijuana can be found by dividing the sum of subjects who tested negative and subjects who used marijuana (x + y) by the total number of subjects (100).

Probability = (x + y) / 100

Since x + y = 100, the probability of the subject tested negative or used marijuana is:

Probability = 100 / 100 = 1

Therefore, the probability that the subject tested negative or used marijuana is 1, or 100%.

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For any m x n matrix A, AAT and ATA are symmetric matrices as  (AAT)^T = AAT and (ATA)^T = ATA.

To prove or disprove that for any m x n matrix A, AAT and ATA are symmetric, we can use the definition of a symmetric matrix and the properties of matrix transposes.
A matrix B is symmetric if B = B^T, where B^T is the transpose of B. So, we need to show that (AAT)^T = AAT and (ATA)^T = ATA.
Step 1: Find the transpose of AAT:
(AAT)^T = (AT)^T * A^T (by the reverse order property of transposes)
(AAT)^T = A * A^T (since (A^T)^T = A)
(AAT)^T = AAT
Step 2: Find the transpose of ATA:
(ATA)^T = A^T * (AT)^T (by the reverse order property of transposes)
(ATA)^T = A^T * A (since (A^T)^T = A)
(ATA)^T = ATA

Since (AAT)^T = AAT and (ATA)^T = ATA, we have proved that for any m x n matrix A, AAT and ATA are symmetric matrices.

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δy = 0.1192 and dy = -1.08.

To find δy and dy, we can use the formula:
δy = f(x+δx) - f(x)
dy = f'(x) * dx
where f(x) = x^4 and x = -3.

First, let's find δy:
δy = f(-3+0.01) - f(-3)
δy = (-3.01)^4 - (-3)^4
δy = 0.1192

Next, let's find dy:
dy = f'(-3) * 0.01
dy = 4(-3)^3 * 0.01
dy = -1.08

Comparing δy and dy, we can see that they have different signs and magnitudes. δy is positive and has a magnitude of 0.1192, while dy is negative and has a magnitude of 1.08.

This makes sense because δy represents the change in y due to a small change in x, while dy represents the instantaneous rate of change of y with respect to x at a specific point.

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The given equation is rearranged to obtain 9x - 1 - (41 - 5x) = 0. By simplifying and solving the equation, we get x = 3.

To solve this equation, we first simplify it by subtracting what is to the right of the equal sign from both sides of the equation. This gives us 9x - 1 - 41 + 5x = 0, which simplifies to 14x - 42 = 0. We then pull out the like factor of 14 to obtain 14(x - 3) = 0.

Next, we use the zero product property to find the values of x that make the equation true. We know that 14 can never equal 0, so we focus on the expression inside the parentheses. Setting x - 3 equal to 0, we get x = 3.

Therefore, the solution to the given equation is x = 3.

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Complete Question:

Simplify 9x-1=41-5x.

Answer:

Step-by-step explanation:

therefore favourable/total outcomes

hope it helps

Answer:

P=70

Step-by-step explanation:

because when you divide y by x you get 70

140 divided by 2 is 70.

Answer:

**NEED ANSWER ASAP DUE TOMORROW**

What are two observations of quasars that prove they cannot be a part of the milky way galaxy?

Step-by-step explanation:

The volume of the solid in the first octant bounded by the parabolic cylinder z=25-x² and the plane y=1 is 32.33 cubic units (approximately).

To find the volume of the solid, we need to integrate the cross-sectional area of the solid with respect to x. Since the plane y=1 cuts the solid into two halves, we can integrate over the range 0 ≤ x ≤ 5.

For a fixed value of x, the cross-sectional area of the solid is given by the area of the ellipse formed by the intersection of the parabolic cylinder and the plane y=1. The equation of this ellipse can be obtained by substituting y=1 in the equation of the parabolic cylinder:

Therefore, the equation of the ellipse is:

We can now integrate the cross-sectional area over the range 0 ≤ x ≤ 5:

where A(x) is the area of the ellipse given by:

Evaluating the integral, we get:

Therefore, the volume of the solid in the first octant bounded by the parabolic cylinder z=25-x² and the plane y=1 is approximately 32.33 cubic units.

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Answer:

the equation is

g=f+10 g=f-10

The value of (f^(-1))'(5) is 1/9, where f(x) = x^3 + 9x + 5.

To find the derivative of the inverse function at a particular point, we can use the formula:

(f^(-1))'(y) = 1 / f'(f^(-1)(y))

where y is the value at which we want to find the derivative of the inverse function.

In this case, we want to find (f^(-1))'(5), so we need to first find f'(x) and f^(-1)(5).

Starting with f'(x):

f(x) = x^3 + 9x + 5

Taking the derivative with respect to x, we get:

f'(x) = 3x^2 + 9

Now we need to find f^(-1)(5):

f(x) = 5

x^3 + 9x + 5 = 5

x^3 + 9x = 0

x(x^2 + 9) = 0

x = 0 or x = ±√(-9)

Since the inverse function is a function, it can only take one value for each input value, so we can discard the solution x = -√(-9), which is not a real number. Therefore, f^(-1)(5) = 0.

Now we can use the formula to find (f^(-1))'(5):

(f^(-1))'(5) = 1 / f'(f^(-1)(5)) = 1 / f'(0)

Substituting into the expression for f'(x), we get:

f'(x) = 3x^2 + 9

f'(0) = 3(0)^2 + 9 = 9

(f^(-1))'(5) = 1 / f'(f^(-1)(5)) = 1 / f'(0) = 1 / 9

So, (f^(-1))'(5) = 1/9

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There are 24 possible four-letter sequences using the letters b, r, j, and w once each.

To find out how many four-letter sequences are possible using the letters b, r, j, and w once each, we can use the formula for permutations of n objects taken r at a time, which is:

P(n,r) = n! / (n-r)!

In this case, n = 4 (since there are 4 letters to choose from) and r = 4 (since we want to choose all 4 letters). So we can plug in these values and simplify:

P(4,4) = 4! / (4-4)!
P(4,4) = 4! / 0!
P(4,4) = 4 x 3 x 2 x 1 / 1
P(4,4) = 24

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Answer:

mean ( averege) is the sum of the value divided by the number of value

mean= (5+12+17+7+7) / 5

=48/5

= 9.6

Answer:

The mean is the sum of the value divided by the number of the values.

i.e. mean = (5+12+17+7+7) / 5

=48/5

= 9.6

The values of the remaining five trigonometric functions are:

cos(x) = 4/5
tan(x) = 3/4
csc(x) = 5/3
sec(x) = 5/4
cot(x) = 4/3.

Using the given information, we can use the Pythagorean identity to find the value of cos(x):

cos²(x) = 1 - sin²(x)
cos²(x) = 1 - (3/5)²
cos²(x) = 1 - 9/25
cos²(x) = 16/25
cos(x) = ±4/5

Since 0 < x < π/2, we know that cos(x) is positive, so cos(x) = 4/5.

Now we can use the definitions of the remaining trigonometric functions to find their values:

tan(x) = sin(x) / cos(x) = (3/5) / (4/5) = 3/4
csc(x) = 1 / sin(x) = 5/3
sec(x) = 1 / cos(x) = 5/4
cot(x) = 1 / tan(x) = 4/3

Therefore, the values of the remaining five trigonometric functions are:

cos(x) = 4/5
tan(x) = 3/4
csc(x) = 5/3
sec(x) = 5/4
cot(x) = 4/3.

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x+y=10

225x+70y=1165 where x represents the quantity of sliders in the combo.

The quantity of chicken wings in the combo is given by y.

An equation is an expression of one or more variables equated to a constant or another expression. Equations are used in mathematics to describe the relationship between two or more variables and often involve operations such as addition, subtraction, multiplication, division, and exponents. Equations can also be used to describe physical phenomena and other phenomena such as chemical reactions, motion, and electricity.
Let x represent the number of sliders in the combination meal and let y represent the number of chicken wings in the combination meal. The system of equations used to determine the number of sliders and chicken wings in the combination meal is given by:
Equation 1: 225x + 70y = 1165
Equation 2: x + y = 10
This system of equations can be used to find the number of sliders and chicken wings in the combination meal. To solve this system, the equations must be manipulated to isolate one of the variables. For example, if we subtract 70y from both sides of Equation 1, we obtain 225x = 1165 - 70y. We can then divide both sides by 225 to obtain x = (1165 - 70y)/225. This expression can then be substituted into Equation 2 to obtain a single equation in one variable.
Therefore, the system of equations given by Equation 1 and Equation 2 can be used to determine the number of sliders and chicken wings in the combination meal.
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x+y=10

225x+70y=1165 where x represents the quantity of sliders in the combo.

The quantity of chicken wings in the combo is given by y.

An equation is an expression of one or more variables equated to a constant or another expression. Equations are used in mathematics to describe the relationship between two or more variables and often involve operations such as addition, subtraction, multiplication, division, and exponents. Equations can also be used to describe physical phenomena and other phenomena such as chemical reactions, motion, and electricity.
Let x represent the number of sliders in the combination meal and let y represent the number of chicken wings in the combination meal. The system of equations used to determine the number of sliders and chicken wings in the combination meal is given by:
Equation 1: 225x + 70y = 1165
Equation 2: x + y = 10
This system of equations can be used to find the number of sliders and chicken wings in the combination meal. To solve this system, the equations must be manipulated to isolate one of the variables. For example, if we subtract 70y from both sides of Equation 1, we obtain 225x = 1165 - 70y. We can then divide both sides by 225 to obtain x = (1165 - 70y)/225. This expression can then be substituted into Equation 2 to obtain a single equation in one variable.
Therefore, the system of equations given by Equation 1 and Equation 2 can be used to determine the number of sliders and chicken wings in the combination meal.
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To compute the area of the transformed ellipse T(S), A transformed ellipse is an ellipse that has been subjected to a transformation, which can change its size, shape, and orientation.

We'll follow these steps:
1. Identify the original ellipse S and its area.
2. Apply the transformation T(x) = Ax.
3. Compute the determinant of matrix A.
4. Compute the area of the transformed ellipse T(S) using the determinant.

Step 1: The original ellipse S has an area of 11.
Step 2: The transformation T(x) = Ax is given by the matrix A = [1 2; 0 -6].
Step 3: Compute the determinant of matrix A.
The determinant is given by det(A) = (1 * -6) - (2 * 0) = -6.
Step 4: Compute the area of the transformed ellipse T(S) using the determinant.
The area of the transformed ellipse T(S) can be computed using the formula:

Area(T(S)) = |det(A)| * Area(S).
Area(T(S)) = |-6| * 11 = 6 * 11 = 66.

The area of the transformed ellipse T(S) is 66.

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The function f is not a function from r to r since none of the given functions are functions from r to r due to either being undefined for certain inputs or having multiple outputs for a single input.

a) f(x) = 1/x

f is not a function from R to R because it is undefined when x = 0. For f to be a function from R to R, it must have a defined output for every real number input.

b) f(x) = √x

f is not a function from R to R because the square root is not defined for negative numbers. Thus, it doesn't have an output for negative real number inputs.

c) f(x) = ±√(x² + 1)

f is not a function from R to R because it violates the definition of a function, which states that each input must have exactly one output. In this case, each input x has two outputs: the positive and negative square roots of (x² + 1).

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Answer:

The function represents a growth. The percentage rate is 9%.

Step-by-step explanation:

f(x)= a(1+r)^t

f(x) 7900 (1+.09)^x

Using the above equations, since the rate is greater than 1, that means the function represents growth. And to find the percentage rate, take the .09 and multiply it by 100 to convert it into a percentage.

(a) the set of all products in which the exponents sum to 4 is (1 + x + x²)4.

(b) the set of all products in which the exponents sum to 4 is (1 + x + x² + x³ + x⁴)4.

(c) the set of all products in which the exponents sum to 4 is (1 + x³ + x⁴)2(1 + x + x²)2.

(d) the set of all products in which the exponents sum to 4 is (1 + x + x² + x³ ......)3.

An exponent is written as a superscript to the right of the base number and indicates the number of times the base number is multiplied by itself.

(a) The set of all formal products in which the exponents sum to 4 can be determined by the following equation: (1 + x + x²)4.

This is because the exponents of each term must sum to 4, thus the exponents of the terms in the product must also sum to 4. Thus, the set of all products in which the exponents sum to 4 is (1 + x + x²)4.

(b) The set of all formal products in which the exponents sum to 4 can be determined by the following equation: (1 + x + x² + x³ + x⁴)4.

This is because the exponents of each term must sum to 4, thus the exponents of the terms in the product must also sum to 4. Thus, the set of all products in which the exponents sum to 4 is (1 + x + x² + x³ + x⁴)4.

(c) The set of all formal products in which the exponents sum to 4 can be determined by the following equation: (1 + x³ + x⁴)2(1 + x + x²)2.

This is because the exponents of each term must sum to 4, thus the exponents of the terms in the product must also sum to 4. Thus, the set of all products in which the exponents sum to 4 is (1 + x³ + x⁴)2(1 + x + x²)2.

(d) The set of all formal products in which the exponents sum to 4 can be determined by the following equation: (1 + x + x² + x³  ......)3.

This is because the exponents of each term must sum to 4, thus the exponents of the terms in the product must also sum to 4. Thus, the set of all products in which the exponents sum to 4 is (1 + x + x² + x³ ......)3.

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The price per cup of the 2-gallon container of laundry detergent costing $30.40 is $0.30 per cup.

To calculate the price per cup, we first need to convert 2 gallons to cups.

1 gallon = 16 cups

So, 2 gallons = 2 x 16 = 32 cups

Now, to find the price per cup, we divide the total price of the container by the number of cups in it:

Therefore, the price per cup of laundry detergent is $0.30 per cup (rounded to two decimal places).

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To find the area of the shaded region, we need to subtract the area of the square from the area of the circle:

Area of shaded region = Area of circle - Area of square

Area of shaded region = 81.3 - 16.1

Area of shaded region = 65.2 cm^2

The error interval for the area, a, of the shaded region can be found by considering the errors in the measurements of the areas of the circle and the square. For the area of the circle, the value is rounded to 1 decimal place, so the error is at most 0.05 cm^2 (half of the value of the smallest decimal place). For the area of the square, the value is truncated to 1 decimal place, so the error is at most 0.1 cm^2 (the value of the smallest decimal place).

Thus, the error interval for the area, a, of the shaded region is:

m < a < n

where:

m = 81.3 - 16.1 - 0.1 = 65.1 cm^2

n = 81.3 - 16.1 + 0.05 = 65.25 cm^2

Therefore, the error interval for the area, a, of the shaded region is:

65.1 < a < 65.25

(a) The probability that a student does not drink once a month= 0.40

(b) The probability corresponding to each outcome would depend on the           given relative  probability

(c) You can calculate the value of this expression to find the probability    that at least one person drinks once a month.

a. The probability that a student does not drink once a month can be calculated as 1 minus the probability that a student drinks once a month, which is given as P(D) = 0.60.

So, P(not D) = 1 - P(D) = 1 - 0.60 = 0.40.

b. If we randomly select 2 students at a time and record their drinking habits, the sample space corresponding to this experiment would consist of all possible combinations of drinking habits for the two students.

Since each student can either drink once a month (D) or not drink once a month (not D), there are four possible outcomes:

Both students drink once a month (D, D)

Both students do not drink once a month (not D, not D)

The first student drinks once a month and the second student does not (D, not D)

The first student does not drink once a month and the second student drinks once a month (not D, D)

The probability corresponding to each outcome would depend on the given relative frequency or probability of a student drinking once a month (P(D)) and the complement of that probability (1 - P(D)).

c. Assuming that 5 students are selected at random, the probability that at least one person drinks once a month can be calculated using the complement rule.

The complement of the event "at least one person drinks once a month" is "none of the 5 students drink once a month" or "all 5 students do not drink once a month".

The probability that one student does not drink once a month is P(not D) = 0.40 (as calculated in part a).

The probability that all 5 students do not drink once a month is [tex](P(not D))^5[/tex], since the events are assumed to be independent.

So, the probability that at least one person drinks once a month is:

P(at least one person drinks once a month) = 1 - P(none of the 5 students drink once a month)

= 1 - [tex](P(not D))^5[/tex]

= 1 - [tex](0.40)^5[/tex] (after substituting the value of P(not D) from part a)

You can calculate the value of this expression to find the probability that at least one person drinks once a month.

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