A line graph titled Car Mileage for a Hybrid Car has number of gallons on the x-axis, and number of miles on the y-axis. 1 Gallon is 60 miles, 2 gallons is 120 miles, 3 gallons is 180 miles, and 4 gallons is 240 miles.
What is the value of y when the value of x is 1?

Ok I’ll tell you how to do it.

1. Base x Hight x length
I can’t solve it because I can’t see all the numbers.

The equations which satisfy (6, -3) are: y = -5x + 27 and y = 2x - 15 (Option C and D)

To know which equation will result in (6, -3), we shall determine the value of y in each equation since we know that x = 6. Details below:

For A

y = -3x + 6

y = -3(6) + 6

y = -18 + 6

y = 12

For B

y = 2x - 9

y = 2(6) - 9

y = 12 - 9

y = 3

For C

y = -5x + 27

y = -5(6) + 27

y = -30 + 27

y = -3

For D

y = 2x - 15

y = 2(6) - 15

y = 12 - 15

y = -3

For E

y = -4x + 27

y = -4(6) + 27

y = -24 + 27

y = 3

From the above, the equation that satisfy (6, -3) are:

Thus, the correct answer to the question is Option C and D

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The slope of the line representing the rope is -1/4.

To find the slope of the line representing the rope, we need to use the formula for slope, which is:

slope = rise / run

In this case, the rise is the height of the wall, which is 6 feet, and the run is the horizontal distance between the wall and the point where the rope is attached to the ground, which is 24 feet. However, we need to be careful about the sign of the slope, since the rope is going down from the wall to the ground.

To take this into account, we can use the convention that a positive slope means a line is going up from left to right, and a negative slope means a line is going down from left to right. In this case, since the rope is going down, we know the slope will be negative.

So, we can calculate the slope as follows:

slope = - rise / run
slope = - 6 / 24
slope = - 1/4

Therefore, the line representing the rope has a slope of -1/4.

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The purchase price of the T-Bill is $1,239.17.

To calculate the purchase price of the T-Bill, we need to use the formula:

Purchase Price = Face Value / (1 + (interest rate x days to maturity / 365))

In this case, the face value is $1260, the interest rate is 5.07% p.a., and the days to maturity is 91. Plugging in the values, we get:

Purchase Price = $1260 / (1 + (0.0507 x 91 / 365))
Purchase Price = $1260 / 1.0123
Purchase Price = $1,239.17 (rounded to the nearest cent)

The purchase price of the T-Bill is $1,239.17.

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a) The coordinates of K' after the translation are given as follows: K'(-2,3).

b) The coordinates of M' after the translation are given as follows: M'(-4, 1).

The four translation rules are defined as follows:

From the vector, the composite translation rule in this problem is given as follows:

(x,y) -> (x - 6, y + 2).

The coordinates of K and M are given as follows:

K(4,1), M(2,-1).

Hence the coordinates of the translated vertices are obtained applying the operation as follows:

K'(-2, 3) and M'(-4, 1).

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The angle measure to the nearest degree of tan B = .5543 is 29°.

Given that tan B = 0 .5543, we need to find the measure to the nearest degree of tan B,

Since, we need to find the measurement of the angle, so we will use the concept of inverse of trigonometric functions,

tan B = 0 .5543

B = tan⁻¹ (0.5543)

B = 28.99 ≈ 29°

Hence, the angle measure to the nearest degree of tan B = .5543 is 29°.

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Among the given numbers, -0.52 is closer to -1/2, 0.35 is closer to 0, and 0.25 is closer to both 0 and 1/2.

To determine which number is closer to -1/2, 0, and 1/2, we need to find the absolute value of the difference between each number and the three given values, and then compare the results.

- For 0.35: The absolute value of the difference between 0.35 and -1/2 is approximately 0.85, the absolute value of the difference between 0.35 and 0 is 0.35, and the absolute value of the difference between 0.35 and 1/2 is approximately 0.15. Therefore, 0.35 is closer to 0 than to -1/2 or 1/2.

- For -3/5: The absolute value of the difference between -3/5 and -1/2 is approximately 0.05, the absolute value of the difference between -3/5 and 0 is approximately 0.6, and the absolute value of the difference between -3/5 and 1/2 is approximately 0.95. Therefore, -3/5 is closer to 0 than to -1/2 or 1/2.

- For -0.52: The absolute value of the difference between -0.52 and -1/2 is approximately 0.02, the absolute value of the difference between -0.52 and 0 is approximately 0.52, and the absolute value of the difference between -0.52 and 1/2 is approximately 1.02. Therefore, -0.52 is closer to -1/2 than to 0 or 1/2.

- For 0.25: The absolute value of the difference between 0.25 and -1/2 is approximately 0.75, the absolute value of the difference between 0.25 and 0 is 0.25, and the absolute value of the difference between 0.25 and 1/2 is approximately 0.25. Therefore, 0.25 is closer to 0 and 1/2 than to -1/2.

- For 3/5: The absolute value of the difference between 3/5 and -1/2 is approximately 1.15, the absolute value of the difference between 3/5 and 0 is approximately 0.6, and the absolute value of the difference between 3/5 and 1/2 is approximately 0.05. Therefore, 3/5 is closer to 0 than to -1/2 or 1/2.

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{u, v, w} is linearly independent, and the statement is proved.

What is linear function?

A linear function is a mathematical function of the form f(x) = mx + b, where m and b are constants.

To prove this statement, we will use a proof by contradiction.

Assume that A is linearly dependent and that {u, v, w} is also linearly dependent. This means that there exist scalars α, β, and γ, not all zero, such that αu + βv + γw = 0.

Now, we will express w in terms of u and v:

w = u + 2v - tw1

Substituting w in the above equation, we get:

αu + βv + γ(u + 2v - tw) = 0

Simplifying this equation, we get:

(α + γ)u + (β + 2γ)v - γtw = 0

Since u, v, and w are distinct and none of them is the zerovector, α + γ ≠ 0 or β + 2γ ≠ 0 or -γt ≠ 0.

If -γt = 0, then γ = 0, which contradicts the assumption that not all scalars α, β, and γ are zero.

If -γt ≠ 0, then we can express t as t = γ' / (-γ) for some non-zero scalar γ'. Substituting t in the above equation, we get:

(α + γ)u + (β + 2γ)v + γ'w = 0

We can now express w in terms of u and v using the equation w = u + 2v - tw1, which gives:

(α + γ)u + (β + 2γ)v + γ'(u + 2v - tw) = 0

Simplifying this equation, we get:

(α + γ + γ')u + (β + 2γ + 2γ')v - γ'tw1 = 0

Since u, v, and w are distinct and none of them is the zerovector, it follows that the coefficients of u, v, and w1 are not all zero. Therefore, {u1, v1, w1} is linearly independent.

This contradicts the assumption that {u, v, w} is linearly dependent. Therefore, our initial assumption that {u, v, w} is linearly dependent must be false. Hence, {u, v, w} is linearly independent, and the statement is proved.

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Given the function h(x) = 2x - 3 and g(x) = -2x + 2, g(h(x)) = -4x + 8

A function is a mathematical relationship that shows the relationship between two variables.

Given the functions h(x) = 2x - 3 and g(x) = -2x + 2

We desire to find (goh)(x), we proceed as follows.

Since we have the functions h(x) = 2x - 3 and g(x) = -2x + 2

We notice that  (goh)(x) = g(h(x))

So, substituting the values of the variables into the equation, we have that

(goh)(x) = g(h(x))

= -2x + 2

Substituting h(x) = 2x - 3 into the equation, we have that

g(h(x)) = -2x + 2

= -2(2x - 3) + 2

= -4x + 6 + 2

= -4x + 8

So, g(h(x)) = -4x + 8

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(a) The algorithm fails when the initial choices of w and h are not of full rank.

(b) The entries of h and w may not always be non-negative, but the algorithm aims for non-negative matrix factorisation.

(c) The algorithm is used to compute a nonnegative matrix factorisation of A = [6 8] using iterative updates of w and h.

(a) An example of a situation where the algorithm would fail is when the initial choices of w and h are not of full rank. In this case, the iterative computation of h and w would not converge to the rank r factorisation of matrix A.

(b) If the algorithm does not fail and the iterative computation of h and w converges to the rank r factorisation of matrix A, then the entries of h and w may not always be non-negative. However, the algorithm is designed to find a non-negative matrix factorisation, so it is expected that the entries of h and w will be non-negative in most cases.

(c) Using the given algorithm, we can compute the rank r nonnegative matrix factorisation of matrix A as follows:

- Set w to be a 2x1 matrix of random non-negative values, and h to be a 1x2 matrix of random non-negative values, both of full rank.
- Compute h = h .* (w' * A) ./ (w' * w * h) and w = w .* (A * h') ./ (w * h * h'), where .* denotes element-wise multiplication, and ' denotes matrix transpose.
- Repeat step 2 until convergence is achieved, or a maximum number of iterations is reached.

Using this algorithm, we can compute the nonnegative matrix factorisation of matrix A as:

w = [0.1829; 0.9119]
h = [3.6953 4.9237]

where w and h are non-negative matrices of rank 1 that satisfy A = w * h.

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Therefore, the area of the trapezoid is 1338 3/4 square units.

A quadrilateral with at least one set of parallel sides is known in geometry as a trapezium, or trapezium in British and other versions of English. In Euclidean geometry, a trapezium is invariably a convex quadrilateral. The trapezoid's parallel sides are referred to as its bases.

To find the area of a trapezoid, we use the formula:

Area = (1/2) × (sum of parallel sides) × (height)

In this case, we have the following information:

The two parallel sides are 12 1/4 and 24 1/4.

The height is 73 1/2.

First, we need to add the two parallel sides to find the sum:

12 1/4 + 24 1/4 = 36 1/2

Next, we can plug these values into the formula:

Area = (1/2) × 36 1/2 × 73 1/2

Area = 1338.75

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Correct Question:

Given sides are 12(1/4), 24 (1/4), 73 (1/2) and 134(1/2), then What is the area of this trapezoid?

Answer:

47 ÷ 3681 is approximately 0.0128

Answer:

The nswer is 0.0127682694919858

The angle of elevation of Elea's line of sight to the top of the tree is 12.4°

The angle of elevation is an angle that is formed between the horizontal line and the line of sight.

The vertical distance or height from the line of sight to the top of the tree is

23.5 - 1.6

= 21.9

This means the height will be the opposite to the angle of elevation and the distance between Elea and the tree is the adjascent.

Using trigonometry;

If tetha is the angle of elevation

tan(tetha) = opp/adj

tan(tetha) = 21.9/100

tan(tetha) = 0.219

tetha = tan^-1( 0.219)

= 12.4°

therefore the angle of elevation is 12.4°

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When x = 2, f(g(x)) is approximately equal to 187/9.

To solve for f(g(x)) when x = 2, we need to substitute the value of x into the function g(x) and then substitute the result into the function f(x). Let's calculate it step by step:

Step 1: Calculate g(x) when x = 2:

g(x) = x + 4/3

g(2) = 2 + 4/3

g(2) = 2 + 4/3

g(2) = 10/3

Step 2: Substitute the result from step 1 into f(x):

f(x) =[tex]x^2[/tex] + 2x + 3

f(g(x)) = f(10/3)

f(g(2)) = f(10/3)

Step 3: Calculate f(g(2)):

f(10/3) = (10/3[tex])^2[/tex] + 2(10/3) + 3

f(10/3) = 100/9 + 20/3 + 3

f(10/3) = 100/9 + 60/9 + 27/9

f(10/3) = 187/9

Therefore, when x = 2, f(g(x)) is approximately equal to 187/9.

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The function f(x) = 5.1 log(x) is a dilation of the parent function by a factor of 5.1

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

f(x) = log x

f'(x) = 5.1 log x

When the above functions are compared, we have

f'(x) = 5.1 log(x)

This means that the function f(x) is dilated by 5.1 to get the function f'(x)

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Two angles are co-terminal, meaning they differ by a multiple of 360°.

a) We know that sin A = opposite/hypotenuse = 5/12. Therefore, the adjacent side of angle A must be negative, since it is located in the second quadrant. We can use the Pythagorean theorem to find the hypotenuse:

(5/12)^2 + (adjacent)^2 = hypotenuse^2

25/144 + (adjacent)^2 = hypotenuse^2

(adjacent)^2 = hypotenuse^2 - 25/144

(adjacent)^2 = (hypotenuse^2 * 144 - 25)/144

We also know that cosine is adjacent/hypotenuse and tangent is opposite/adjacent, so:

cos A = adjacent/hypotenuse = sqrt(hypotenuse^2 - 25/144)/hypotenuse

tan A = opposite/adjacent = 5/sqrt(hypotenuse^2 - 25/144)

b) To find angle A, we can use the inverse sine function:

A = sin^-1(5/12)

A ≈ 24.02°

We know that cosecant is the reciprocal of sine, so:

csc A = 1/sin A

We want to find angles that have a cosecant of V3, so:

1/sin A = V3

sin A = 1/V3

We can use the unit circle to find angles whose sine is 1/V3. One such angle is 60°, since sin 60° = V3/2. Another angle is 300°, since sin 300° = -V3/2. These two angles are co-terminal, meaning they differ by a multiple of 360°.

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a.  The probability distribution of the test statistic is approximately a standard normal distribution.

b. The p-value for the test of  factors that affect the prevalence of tuberculosis among intravenous drug users is 0.0202.  

c. We can conclude that there is a statistically significant difference between the two groups in terms of their proportions of positive skin test results.

d. The 95% confidence interval does not contain zero, so there is a statistically significant difference between the two groups in terms of their proportions of positive skin test results.

To estimate the common value of p assuming that the population proportions of positive skin test results are equal, we can compute the pooled proportion:

p-hat = (x1 + x2) / (n1 + n2)

= (24.7 + 17.4) / (97 + 161)

= 0.195

where x1 and x2 are the number of individuals with positive skin test results in the two groups, and n1 and n2 are the sample sizes.

a. To test the null hypothesis that the proportions of intravenous drug users who have positive tuberculin skin test results are identical for those who share needles and those who do not, we can use a two-sample z-test for proportions. The test statistic is:

z = (p1 - p2) / sqrt(phat * (1 - phat) * (1/n1 + 1/n2))

where p1 and p2 are the sample proportions, phat is the pooled proportion, and n1 and n2 are the sample sizes.

Plugging in the values, we get:

z = (0.247 - 0.174) / sqrt(0.195 * (1 - 0.195) * (1/97 + 1/161))

= 2.05

The probability distribution of the test statistic is approximately a standard normal distribution, since the sample sizes are large enough (both n1 and n2 are greater than 30).

b. The p-value for the test is the probability of observing a z-value of 2.05 or more extreme under the null hypothesis. From a standard normal distribution table or calculator, we find that the p-value is approximately 0.0202 (or 0.0404 for a two-tailed test).

Since the p-value is less than the significance level of 0.05, we reject the null hypothesis and conclude that the proportions of intravenous drug users who have positive tuberculin skin test results are not identical for those who share needles and those who do not.

c. To construct a 95% confidence interval for the true difference in proportions, we can use the formula:

(p1 - p2) ± z* sqrt(phat * (1 - phat) * (1/n1 + 1/n2))

where z is the critical value for a 95% confidence interval from a standard normal distribution (z = 1.96).

Plugging in the values, we get:

(0.247 - 0.174) ± 1.96 * sqrt(0.195 * (1 - 0.195) * (1/97 + 1/161))

= 0.073 ± 0.090

Therefore, we can be 95% confident that the true difference in proportions of intravenous drug users who have positive tuberculin skin test results between those who share needles and those who do not is between 0.073 and -0.073 (which can be written as an absolute value of 0.073).

d.  We can infer that there is a statistically significant difference between the two groups in terms of the proportions of positive skin test results because the interval does not contain zero.

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The frog's probability of being on lily pad B after an hour (or in the long run) is around 0.2975 or 29.75%.

To calculate the probability that the frog will be on lily pad B after an hour, we can start by calculating the long-run distribution or steady-state probabilities of the frog's movement between the two lily pads.

Let p represent the probability that the frog is on lily pad B at any given time. We may develop two equations based on the probabilities given in the problem:

p = 0.85(1-p) (the frog moves from point A to point B with a probability of 0.85 and stays on point B with a probability of 1-p).

1-p = 0.65p (the frog moves from B to A with the probability of 0.65 and stays on A with the probability of 1-p).

Simplifying the equation, we get:

p = 0.85 - 0.85p + 0.65p

p = 0.85(1 - 0.65) = 0.2975

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Answer: #3 = 22/50 (reduced version is 11/25

Step-by-step explanation:

Answer:

Step-by-step explanation:

Predicted probabilities are different then experimental probabilities.  Experimental probabilities use the actual data.

P(red)= red/(total)= 22/(12+15+22)  = 22/50 = .44

P(hot cocoa = hot cocoa/total =5/(7+5+8) = 1/4 = .25

Louisa would need to invest $32,172.75 semiannually to have $44000 in four years if the fund she has selected pays 8.1% annually, compounded semiannually.

To have $44000 in four years, Louisa would need to invest a total of $x in the fund that pays 8.1% annually, compounded semiannually.

Using the formula for compound interest, we can solve for x:

A = P(1 + r/n)^(nt)

Where:
A = the total amount (in this case, $44000)
P = the principal amount (the amount Louisa needs to invest)
r = the annual interest rate (8.1%)
n = the number of times the interest is compounded per year (semiannually = 2)
t = the number of years (4)

Plugging in these values:

44000 = P(1 + 0.081/2)^(2*4)

44000 = P(1.0405)^8

44000 = P(1.366)

P = 32172.75

So Louisa would need to invest $32,172.75 semiannually to have $44000 in four years if the fund she has selected pays 8.1% annually, compounded semiannually.

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Question: "Identify the correct description for the formula g'(x) ≈ g(x)/h – g(x – h)/h from the following options: FFD1: forward finite difference with stepsize h for the first derivative of g at a BFD1: backward finite difference with stepsize h for the first derivative of g at a CFD1: central finite difference with stepsize h for the first derivative of g at x CFD2: central finite difference with stepsize h for the second derivative of g at x None of the Above"

The correct description for the formula g'(x) ≈ g(x)/h – g(x – h)/h is BFD1: backward finite difference with stepsize h for the first derivative of g at a.

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The general solution is y = [tex][1/(-1/2x^2 - 3/2x^6 + c + K)]^_{(1/3)[/tex], where c and K are arbitrary constants.

To tackle the differential condition [tex]y^3 - (18x^8)3xy^2y' = 0[/tex], we can utilize detachment of factors.

In the first place, we can improve the condition to get: [tex]y^2y' = (y/x)^3 - 18x^5[/tex].

Then, we can isolate the factors by duplicating the two sides by dx and partitioning the two sides by [tex](y^2(y/x)^3 - 18x^5)[/tex] to get:

[tex](y^2/y^3)dy = [(1/x)^3 - 18x^3]dx[/tex]

Incorporating the two sides, we get:

[tex]-1/y + c = (- 1/2x^2) - (3/2)x^6 + K[/tex]

Where K is an erratic steady of coordination.

At last, we can settle for y to get:

[tex]y = [1/(- 1/2x^2 - 3/2x^6 + c + K)]^_{(1/3)[/tex]

where c + K is the erratic steady.

Accordingly, the overall answer for the differential condition is:

[tex]y^3 - (18x^8)3xy^2y' = 0[/tex] is [tex]y = [1/(- 1/2x^2 - 3/2x^6 + c + K)]^(1/3)[/tex], where c and K are inconsistent constants.

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1. y = 2
dy/dx = 0 (Constant terms have a derivative of 0)

2. y = 2x^2 + 2
dy/dx = 4x (Apply power rule: d(ax^n)/dx = a * n * x^(n-1))

3. y = 2x
dy/dx = 2 (Linear terms have a derivative equal to their coefficient)

4. y = 4x^3 - 4
dy/dx = 12x^2 (Apply power rule and constant term has derivative 0)

5. y = 3x^6
dy/dx = 18x^5 (Apply power rule)

6. y = 2(x-5)^2
dy/dx = 4(x-5) (Apply chain rule: d(u^2)/dx = 2u * du/dx)

7. y = 1 - 3x
dy/dx = -3 (Linear terms have a derivative equal to their coefficient)

8. y = 2/x^3
dy/dx = -6/x^4 (Rewrite as 2x^(-3) and apply power rule)

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The solution of the linear equation:

7x – 13 = ─2x + 5

is x = 2

Here we want to find the value of x that is a solution of:

7x - 13 = -2x + 5

To solve it, we need to isolate x in one of the sides of the equation.

7x - 13 = -2x + 5

7x + 2x = 5 + 13

9x = 18

x = 18/9

x = 2

The value x = 2 makes the given linear equation true.

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The probability that a customer will use plus gas and fill the tank is 0.21 or 21%.

Let's use the following notation:

R: the event that a customer uses regular gas

P: the event that a customer uses plus gas

M: the event that a customer uses premium gas

F: the event that a customer fills their tank

We are given:

P(R) = 0.4, P(P) = 0.35, P(M) = 0.25

P(F|R) = 0.3, P(F|P) = 0.6, P(F|M) = 0.5

We want to find P(P and F), the probability that a customer uses plus gas and fills their tank. We can use the following formula:

P(P and F) = P(F and P) = P(F|P) * P(P)

Substituting the values, we get:

P(P and F) = P(F|P) * P(P) = 0.6 * 0.35 = 0.21

Therefore, the probability that a customer will use plus gas and fill the tank is 0.21 or 21%.

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The perimeter of the square after the dilation of scale factor of 4/3 is 180 units.

Given that,

Perimeter of the square = 135 units = 4a, where 'a' is the length of a side.

Scale factor = 4/3

We have to find the perimeter of the square if the square is dilated by a scale factor of 4/3.

If the square is dilated by a scale factor of 4/3,

length of each side = 4/3 a

Perimeter of the new square = 4 × 4/3 a

                                                = 4/3 × 4a

                                                = 4/3 × 135

                                                = 180 units

Hence the new perimeter of the square is 180 units.

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Answer:

[tex] {9}^{x} \div {81}^{ \frac{2}{x} } = 1[/tex]

[tex]{9}^{x} \times {81}^{ \frac{ - 2}{x} } = 1[/tex]

[tex]{9}^{x} \times { ({9}^{2}) }^{ \frac{ - 2}{x} } = 1[/tex]

[tex]{9}^{x} \times {9}^{ \frac{ - 4}{x} } = 1[/tex]

[tex] {9}^{(x - \frac{4}{x}) } = 1[/tex]

[tex]{9}^{0 } = 1 \: \: \: \: \: (known)[/tex]

Equating the exponents

[tex]x - \frac{4}{x} = 0[/tex]

[tex] \frac{ {x}^{2} - 4 }{x} = 0[/tex]

[tex] {x}^{2} = 4 \\ x = + 2 \: or \: 2[/tex]

The general solution to the original differential equation is:  

y(t) = [tex]C1 e^t cos t + C2 e^t sin t + (1/2)ex sin t + (1/4)ex sin(2t) + (1/4)ln|[/tex]

We first solve the associated homogeneous differential equation:

[tex]2y'' - 4y' + 4y[/tex] = 0

The characteristic equation is[tex]r^2[/tex] - 2r + 2 = 0, which has roots r = 1 ± i. Therefore, the general solution to the homogeneous equation is:

[tex]y_h(t) = e^t([/tex]C1 cos t + C2 sin t)

To use the method of variation of parameters to find the particular solution to the original equation, we assume that the solution has the form:

[tex]y_p(t) = u(t)e^t cos t + v(t)e^t sin t[/tex]

where u(t) and v(t) are functions to be determined.

[tex]y_p''(t) \\\\2u'(t)e^t cos t + 2v'(t)e^t sin t + 2u(t)e^t cos t - 2v(t)e^t sin t - 2u(t)e^t sin t - 2v(t)e^t cos t[/tex]

[tex]y_p'(t) = u'(t)e^t cos t + v'(t)e^t sin t + u(t)e^t cos t + v(t)e^t sin t[/tex]

Substituting these into the original equation and simplifying, we get:

[tex]2u'(t)e^t cos t + 2v'(t)e^t sin t = ex sec x[/tex]

We need to find u'(t) and v'(t) such that this equation holds for all t. To do this, we take the derivative of the assumed solution with respect to t and equate coefficients of cos t and sin t separately:

[tex]u'(t)e^t cos t + v'(t)e^t sin t + u(t)e^t cos t + v(t)e^t sin t = 0 (1)\\v'(t)e^t cos t - u'(t)e^t sin t + u(t)e^t sin t - v(t)e^t cos t = ex sec x (2)[/tex]

Solving equation (1) for u'(t) and v'(t) and substituting into equation (2), we get:

[tex]v(t) = ∫ [ex sec x / (e^(2t))] dt\\u(t) = -∫ [ex sec x / (e^(2t))] tan t dt[/tex]

Evaluating the integrals, we get:

[tex]v(t) = (1/2)ex tan x - (1/2)ln|cos x| + C1\\u(t) = (1/4)ex [sin(2t) - 2cos(2t)] + (1/4)ln|cos x| tan x + C2[/tex]

where C1 and C2 are arbitrary constants.

The general solution to the original differential equation is:  

y(t) = [tex]C1 e^t cos t + C2 e^t sin t + (1/2)ex sin t + (1/4)ex sin(2t) + (1/4)ln|[/tex]

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The requried coordinates of points S', R', and T' are (5,-4), (8,5), and (3,2).

To find the coordinates of the image points after rotating 90 degrees clockwise, we can use the following formulas:

x' = y

y' = -x

For point S(4,5), the coordinates of the image point S' after rotating 90 degrees clockwise can be found as follows:

x' = y = 5

y' = -x = -4

Therefore, the image point S' is (5,-4).

For point R(-5,8), the coordinates of the image point R' after rotating 90 degrees clockwise can be found as follows:

x' = y = 8

y' = -x = 5

Therefore, the image point R' is (8,5).

For point T(-2,3), the coordinates of the image point T' after rotating 90 degrees clockwise can be found as follows:

x' = y = 3

y' = -x = 2

Therefore, the image point T' is (3,2).

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Here is the function to show how much money Milo saves each week:

f(x) = 0.5(x - 15)

A mathematical function is a rule that relates each element of a set called the domain to exactly one element of a set called the range.

The domain is the set of all possible input values for the function, and the range is the set of all possible output values.

Taking Milo for example,

Milo saves each week, given his weekly allowance x, can be expressed as:

f(x) = 0.5(x - 15)

where

x = amount of money Milo saves each week.

if Milo's weekly allowance is $50, then his savings can be calculated using the function as follows:

f(50) = 0.5(50 - 15) = 0.5(35) = 17.5

Therefore, Milo saves $17.5 each week when his weekly allowance is $50.

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