There are functions of the form x^{r} that solve the differential equation x²y"-6xy' + 10 y=0
Give the solution to the initial value problem [x²y"-6xy' + 10 y=0 y(1)=0 y'(1)=3]

A.  P(x > 19) is also approximately 0.0228.

B. P(x < 13) is also approximately 0.1587.

C. P(15.58 < x < 19.58) is also approximately 0.4893.

D. P(10.28 ≤ x ≤ 17.94) is also approximately 0.8226.

a. P(x>19):

We need to standardize the variable x using the z-score formula:

z = (x - µ) / σ

Substituting the values we get,

z = (19 - 15) / 2 = 2

Using a standard normal distribution table or calculator, we find that P(z > 2) is approximately 0.0228. Therefore, P(x > 19) is also approximately 0.0228.

b. P(x < 13):

Again, we use the z-score formula:

z = (x - µ) / σ

Substituting the values we get,

z = (13 - 15) / 2 = -1

Using a standard normal distribution table or calculator, we find that P(z < -1) is approximately 0.1587. Therefore, P(x < 13) is also approximately 0.1587.

c. P(15.58 < x < 19.58):

We need to standardize both values of x using the z-score formula:

z1 = (15.58 - 15) / 2 = 0.29

z2 = (19.58 - 15) / 2 = 2.29

Using a standard normal distribution table or calculator, we find that P(0 < z < 2.29) is approximately 0.9893 - 0.5 = 0.4893. Therefore, P(15.58 < x < 19.58) is also approximately 0.4893.

d. P(10.28 ≤ x ≤ 17.94):

We standardize both values of x using the z-score formula:

z1 = (10.28 - 15) / 2 = -2.36

z2 = (17.94 - 15) / 2 = 0.97

Using a standard normal distribution table or calculator, we find that P(-2.36 ≤ z ≤ 0.97) is approximately 0.8325 - 0.0099 = 0.8226. Therefore, P(10.28 ≤ x ≤ 17.94) is also approximately 0.8226.

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The company makes a profit of $570 by producing and selling 64 units.Given that the cost of producing x units is given by C(x) = 50x + 310 and revenue from selling x units is determined by the price per unit times the number of units sold, thus R(x) = 60x.

To find and interpret (R - C)(64).

Solution:(R - C)(64) = R(64) - C(64)R(x) = 60x, therefore R(64) = 60(64) = $3840.C(x) = 50x + 310, therefore C(64) = 50(64) + 310 = $3270

Hence, (R - C)(64) = R(64) - C(64) = 3840 - 3270 = $570.

Therefore, the company makes a profit of $570 by producing and selling 64 units.

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To find an expression that represents the difference when (5x + 6y) is subtracted from (2x + 7y), we need to subtract (5x + 6y) from (2x + 7y).

When we subtract (5x + 6y) from (2x + 7y), we get:(2x + 7y) - (5x + 6y) = 2x + 7y - 5x - 6yNow we can simplify the expression by combining like terms. The like terms are the x terms and the y terms, so we group them separately:2x - 5x + 7y - 6y = -3x + ySo the expression that represents the difference when (5x + 6y) is subtracted from (2x + 7y) in simplest terms is: -3x + y.Note: The expression -3x + y represents the difference of the terms 2x + 7y and 5x + 6y.

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Rachel's salary during the fifth year will be P^(8316.15) per month. Rachel gets a starting salary of P^(6000) per month and an increase of p% annually.

We are required to calculate her salary during the fifth year. To calculate the salary during the fifth year, we need to find out the salary for each of the five years. The salary during the first year will be P^(6000), and the salary during the second year can be calculated as follows:

Salary after the first year = P^(6000) + P^(6000) × p/100

= P^(6000) × (1 + p/100)

Similarly, the salary during the third year will be: Salary after the second year = P^(6000) × (1 + p/100) + P^(6000) × (1 + p/100) × p/100

= P^(6000) × (1 + p/100)^2

Similarly, we can calculate the salaries for the fourth and fifth years as: Salary after the third year = P^(6000) × (1 + p/100)^3

Salary after the fourth year = P^(6000) × (1 + p/100)^4

Salary after the fifth year = P^(6000) × (1 + p/100)^5

Given that Rachel gets an increase of p% annually, we can use the compound interest formula to calculate the value of p as follows:

We know that P^(8316.15) = P^(6000) × (1 + p/100)^5

Taking the fifth root on both sides, we get:1 + p/100 = (P^(8316.15) / P^(6000))^(1/5)

Substituting the values, we get:1 + p/100 = (1.3817217)

The value of p can be calculated as follows: p/100 = 0.3817217p = 38.17217%

Thus, Rachel's salary during the fifth year will be P^(8316.15) per month, which is approximately P^(8316).

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The margin of error (E) can be calculated using the formula [tex]E = z_{\frac{\alpha}{2}}\sqrt{\frac{p^*(1-p^*)}{n}}$[/tex], where [tex]z_{\frac{\alpha}{2}}$[/tex] is the z-value with a cumulative probability of -2.33. Using the standard normal distribution table, the z-value corresponding to 0.01 is -2.33. The margin of error (E) is 0.0736, allowing for a 95% confidence interval for the true population proportion (p) using the normal approximation to the binomial distribution.

The formula to calculate the margin of error in this case is given by the formula below: [tex]$E = z_{\frac{\alpha}{2}}\sqrt{\frac{p^*(1-p^*)}{n}}$[/tex],

where [tex]$z_{\frac{\alpha}{2}}$[/tex] is the z-value with a cumulative probability of [tex]$\frac{\alpha}{2}$, $p^*$[/tex]

is the sample proportion, and n is the sample size. Now, given that p^ = 0.8, n = 120 and α = 0.02, we can calculate the margin of error (E) as follows:

Firstly, we need to find the z-value with a cumulative probability of

[tex]$\frac{\alpha}{2}$ or $\frac{0.02}{2}[/tex] = 0.01

in the standard normal distribution table. The z-value corresponding to 0.01 is -2.33. Then, substituting these values into the formula above we get:

[tex]$$E = z_{\frac{\alpha}{2}}\sqrt{\frac{p^*(1-p^*)}{n}} = -2.33\sqrt{\frac{0.8(1-0.8)}{120}}$$ $$E = 0.0736$$[/tex]

Therefore, the margin of error (E) is 0.0736. This means that we can construct a confidence interval for the true population proportion (p) with 95% confidence using the formula below[tex]:$$CI = \left(p^ - E, p^ + E \right)$$[/tex] Where p^ is the sample proportion. Now substituting the values given above we get:[tex]$$CI = \left(0.8 - 0.0736, 0.8 + 0.0736 \right)$$ $$CI = (0.7264, 0.8736)$$[/tex]

Hence, the 95% confidence interval for the true population proportion (p) is (0.7264, 0.8736). We used the normal approximation to the binomial distribution since the sample size is large enough.

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h(x, y) is a Lipschitzian function with Lipschitz constant K = K1 * K2.

a) To prove that h(x, y) = f(x, y) + g(x, y) is a Lipschitzian function, we need to show that there exists a constant K such that for any two points (x1, y1) and (x2, y2), the following inequality holds:

| h(x1, y1) - h(x2, y2) | ≤ K * || (x1, y1) - (x2, y2) ||

where || (x1, y1) - (x2, y2) || represents the Euclidean distance between the points (x1, y1) and (x2, y2).

Since f(x, y) and g(x, y) are Lipschitzian functions, we know that there exist constants K1 and K2 such that:

| f(x1, y1) - f(x2, y2) | ≤ K1 * || (x1, y1) - (x2, y2) ||  ... (1)

| g(x1, y1) - g(x2, y2) | ≤ K2 * || (x1, y1) - (x2, y2) ||  ... (2)

Now, let's consider the difference h(x1, y1) - h(x2, y2):

h(x1, y1) - h(x2, y2) = [f(x1, y1) + g(x1, y1)] - [f(x2, y2) + g(x2, y2)]

                     = [f(x1, y1) - f(x2, y2)] + [g(x1, y1) - g(x2, y2)]

Using the triangle inequality, we have:

| h(x1, y1) - h(x2, y2) | ≤ | f(x1, y1) - f(x2, y2) | + | g(x1, y1) - g(x2, y2) |

Applying inequalities (1) and (2), we get:

| h(x1, y1) - h(x2, y2) | ≤ K1 * || (x1, y1) - (x2, y2) || + K2 * || (x1, y1) - (x2, y2) ||

Since K = K1 + K2, we can rewrite the above inequality as:

| h(x1, y1) - h(x2, y2) | ≤ K * || (x1, y1) - (x2, y2) ||

Therefore, h(x, y) is a Lipschitzian function with Lipschitz constant K.

b) To prove that h(x, y) = f(x, g(x, y)) is a Lipschitzian function, we need to show that there exists a constant K such that for any two points (x1, y1) and (x2, y2), the following inequality holds:

| h(x1, y1) - h(x2, y2) | ≤ K * || (x1, y1) - (x2, y2) ||

Let's consider the difference h(x1, y1) - h(x2, y2):

h(x1, y1) - h(x2, y2) = f(x1, g(x1, y1)) - f(x2, g(x2, y2))

Since f(x, y) is a Lipschitzian function, we know that there exists a constant K1 such that:

|

f(x1, g(x1, y1)) - f(x2, g(x2, y2)) | ≤ K1 * || (x1, g(x1, y1)) - (x2, g(x2, y2)) ||

Now, let's consider the distance || (x1, y1) - (x2, y2) ||:

|| (x1, y1) - (x2, y2) || = || (x1, g(x1, y1)) - (x2, g(x2, y2)) ||

Since g(x, y) is a Lipschitzian function, we know that there exists a constant K2 such that:

|| (x1, g(x1, y1)) - (x2, g(x2, y2)) || ≤ K2 * || (x1, y1) - (x2, y2) ||

Combining these inequalities, we have:

| h(x1, y1) - h(x2, y2) | ≤ K1 * || (x1, g(x1, y1)) - (x2, g(x2, y2)) || ≤ K1 * K2 * || (x1, y1) - (x2, y2) ||

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The graph G can be represented by the shorthand notation G = (V, E), where V is the set of vertices and E is the set of edges.

To write the mathematical description of G4 in terms of (V, E), we need to consider the graph G with four iterations. It can be denoted as G4 = (V4, E4), where V4 is the set of vertices in the fourth iteration and E4 is the set of edges in the fourth iteration.

The adjacency matrix A of graph G represents the connections between vertices. It is a square matrix where the entry A[i][j] is 1 if there is an edge between vertices i and j, and 0 otherwise.

To calculate  [tex]A^2[/tex], we need to multiply the adjacency matrix A with itself. The resulting matrix represents the number of paths of length 2 between vertices.

To find the number of paths of length 2 from vertex 0 to vertex 1, we can look at the entry  [tex]A^2[/tex][0][1]. The value of this entry indicates the number of paths of length 2 from vertex 0 to vertex 1. Similarly, we can determine the number of paths of length 2 from vertex 0 to vertex 2 by examining the entry  [tex]A^2[/tex][0][2].

In summary, the shorthand notation for the graph G is G = (V, E). The mathematical description of G4 is G4 = (V4, E4). The adjacency matrix A represents the connections between vertices in G. To calculate [tex]A^2[/tex], we multiply A with itself. The number of paths of length 2 from vertex 0 to vertex 1 is determined by the entry  [tex]A^2[/tex][0][1], and the number of paths of length 2 from vertex 0 to vertex 2 is determined by the entry [tex]A^2[/tex][0][2].

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The equation of the plane passing through the point (0,0,0) with normal vector n=i+j+k is x + y + z = 0

The equation of a plane can be determined when the normal vector and a point on the plane are known. Given that the point (0,0,0) lies on the plane and its normal vector is n = i + j + k, the equation of the plane can be determined as follows:

Step-by-step solution:

Let the equation of the plane be Ax + By + Cz + D = 0

where A, B, C, and D are constants to be determined and (x, y, z) is a point on the plane.

The normal vector of the plane is given as n = i + j + k. This vector is perpendicular to every vector lying on the plane.

Now let's take a point on the plane, say (0, 0, 0).

This vector is parallel to the plane, so its dot product with the normal vector of the plane should be zero.i.e.

0 + 0 + 0 = (0)(1) + (0)(1) + (0)(1)

This gives us: 0 = 0. Hence, the point (0,0,0) satisfies the equation of the plane.

Substituting these values into the equation of the plane, we get:

A(0) + B(0) + C(0) + D = 0

Simplifying, we obtain:

D = 0

Therefore, the equation of the plane is Ax + By + Cz = 0, where A, B, and C are constants to be determined and (x, y, z) is a point on the plane.

Now let's find the values of A, B, and C. To do so, we need to find another point on the plane.

Since the normal vector of the plane is i + j + k, we can choose another point with coordinates that are multiples of the coefficients of this vector. Let's choose the point (1,1,1).

Substituting (1,1,1) into the equation of the plane, we get:

A(1) + B(1) + C(1) = 0

Simplifying, we get:

A + B + C = 0

Therefore, the equation of the plane passing through the point (0,0,0) with normal vector n=i+j+k is x + y + z = 0

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The probability that exactly 11 of 37 randomly selected college students major in STEM can be calculated using the binomial probability formula, which is:

P(X = k) = (n choose k) * p^k * q^(n-k)Where:

P(X = k) is the probability of k successesn is the total number of trials (37 in this case)k is the number of successes (11 in this case)

p is the probability of success (30%, or 0.3, in this case)q is the probability of failure (100% - p, or 0.7, in this case)(n choose k) is the binomial coefficient, which can be calculated using the formula

:(n choose k) = n! / (k! * (n-k)!)where n! is the factorial of n, or the product of all positive integers from 1 to n.

The calculation of the probability of exactly 11 students majoring in STEM is therefore:P(X = 11)

= (37 choose 11) * (0.3)^11 * (0.7)^(37-11)P(X = 11) ≈ 0.200

So the probability that exactly 11 of the 37 randomly selected college students major in STEM is approximately 0.200 or 20%.

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If the formula r= d/t where d is the distance in miles, r is the rate, and t is the time in hours, you can travel at a rate of 75mph to cover 337.5 miles in 4.5 hours.

To calculate at which rate you travel to cover 337.5 miles in 4.5 hours, follow these steps:

Therefore, at a rate of 75 miles per hour, you can travel to cover 337.5 miles in 4.5 hours.

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To evaluate the integral ∫(2+3x)dx, we can use the power rule of integration. However, we need to be careful when handling the absolute value of the expression 2+3x.

Let's first rewrite the expression as U = 2+3x. Now, differentiating both sides with respect to x gives dU = 3dx. Rearranging, we have dx = (1/3)dU.

Substituting these expressions into the original integral, we get ∫(2+3x)dx = ∫U(1/3)dU = (1/3)∫UdU.

Using the power rule of integration, we can integrate U as U^2/2. Thus, the integral becomes (1/3)(U^2/2) + C, where C is the constant of integration.

Finally, substituting back U = 2+3x, we have (1/3)((2+3x)^2/2) + C as the result of the integral.

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To determine whether the expression 6×7N − 2×3N is divisible by 4 for N≥1, let's simplify the expression first:

6×7N − 2×3N = 42N - 6N = 36N.

Now we need to check whether 36N is divisible by 4 for N≥1.

We know that a number is divisible by 4 if its last two digits (in decimal representation) are divisible by 4.

In this case, we are dealing with a variable N, so we need to analyze the possibilities for the last two digits of N that would make 36N divisible by 4.

The last two digits of N can be 00, 01, 02, ..., 98, or 99. Let's consider each case:

1. N = 00: 36N = 36×00 = 0. Divisible by 4.

2. N = 01: 36N = 36×01 = 36. Not divisible by 4.

3. N = 02: 36N = 36×02 = 72. Not divisible by 4.

4. N = 03: 36N = 36×03 = 108. Divisible by 4.

5. N = 04: 36N = 36×04 = 144. Divisible by 4.

6. N = 05: 36N = 36×05 = 180. Divisible by 4.

7. N = 06: 36N = 36×06 = 216. Divisible by 4.

8. N = 07: 36N = 36×07 = 252. Divisible by 4.

9. N = 08: 36N = 36×08 = 288. Divisible by 4.

10. N = 09: 36N = 36×09 = 324. Divisible by 4.

From the analysis above, we can conclude that for N≥1, the expression 6×7N − 2×3N is divisible by 4.

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We cannot determine the percentage of pegilar grade gasoline sales between 3.27 and 53.63 per gallon or the percentage of regular gasoline sale price > 3.33/gallon as the total sales for both are not provided.

Given data:Pegilar grade gasoline sales between 3.27 and 53.63 per gallon.

Percentage of pegilar grade gasoline sale between 3.27 and 53.63 per gallon can be calculated as:X %.

Therefore,X% = (Sale between 3.27 and 53.63 per gallon) / Total sales * 100.

However, the total sales are not provided so we cannot calculate the percentage.

Further information is required.Similarly, for the second part, given data is:Regular gasoline sale price > 3.33/gallon.

Percentage of regular gasoline sale price > $3.33/gallon can be calculated as:Y %.

Therefore,Y % = (Regular sale price > $3.33/gallon) / Total sales * 100.

However, the total sales are not provided so we cannot calculate the percentage. Further information is required.

To summarize, we cannot determine the percentage of pegilar grade gasoline sales between 3.27 and 53.63 per gallon or the percentage of regular gasoline sale price > 3.33/gallon as the total sales for both are not provided.

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The equation of the tangent plane to the surface z = e^(3x/17)ln(4y) at the point (1, 3, 2.96449) is:  z - 2.96449 = (3/17)e^(3/17)(x - 1)ln(4)(y - 3).

To find the equation of the tangent plane, we need to compute the partial derivatives of the given surface with respect to x and y. Let's denote the given surface as f(x, y) = e^(3x/17)ln(4y). The partial derivatives are:

∂f/∂x = (3/17)e^(3x/17)ln(4y), and

∂f/∂y = e^(3x/17)(1/y).

Evaluating these partial derivatives at the point (1, 3), we get:

∂f/∂x (1, 3) = (3/17)e^(3/17)ln(12),

∂f/∂y (1, 3) = e^(3/17)(1/3).

Using these values, we can construct the equation of the tangent plane using the point-normal form:

z - 2.96449 = [(3/17)e^(3/17)ln(12)](x - 1) + [e^(3/17)(1/3)](y - 3).

Simplifying this equation further will yield the final equation of the tangent plane.

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The area bounded by the first quadrant loop of the curve x^5 + y^5 = 3xy is approximately 0.536 square units.

To find the area bounded by the curve x^5 + y^5 = 3xy in the first quadrant, we can use the double integral. However, this particular curve is quite complicated to work with directly. Instead, we can use a change of variables to simplify the equation.

Let's make the substitution u = x^5 and v = y^5. Then, we can express the curve equation in terms of u and v:

u + v = 3uv

This is a much simpler equation to work with. Now, let's find the limits of integration for u and v. Since we are considering the first quadrant, both u and v must be positive. From the original equation, we can see that when x = 0, y = 0, and when y = 0, x = 0. Therefore, the limits of integration for u and v are both from 0 to 1.

Now, we can calculate the area using the double integral:

A = ∬R dA

A = ∫∫R du dv

A = ∫[0,1] ∫[0,1] du dv

A = ∫[0,1] u=0 to 1 v=0 to 1 du dv

A = ∫[0,1] (v/2 + v^2/3) u=0 to 1 dv

A = ∫[0,1] (1/2 + v/3) dv

A = (1/2)v + (1/6)v^2 from 0 to 1

A = (1/2)(1) + (1/6)(1^2) - (1/2)(0) - (1/6)(0^2)

A = 1/2 + 1/6

A = 3/6 + 1/6

A = 4/6

A ≈ 0.667 square units

Therefore, the area bounded by the first quadrant loop of the curve x^5 + y^5 = 3xy is approximately 0.667 square units.

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The difference between 1.42 and 0.53 is 0.37.

The model can be used to determine the difference between 1.42 and 0.53.

First, we start with 1 whole and 4 tenths (1.4) and represent it in the model. Next, we subtract 5 tenths (0.5) from 4 tenths (0.4). Since we cannot subtract directly, we need to regroup. We can regroup 1 whole into 10 tenths and then regroup 1 tenth into 10 hundredths. Now we have 10 tenths (1) and 40 hundredths (0.4).

Next, we subtract 3 hundredths (0.03) from 40 hundredths (0.4). This can be done directly since the place values match. Subtracting, we get 37 hundredths (0.37).

Therefore, the difference between 1.42 and 0.53 is 0.37.

To summarize, we regrouped to subtract 5 tenths from 4 tenths, and then subtracted 3 hundredths from 40 hundredths. The final answer is 0.37.

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The answer is (c) 75. The number of triangles that can be formed using the points A, B, and C as vertices is 1. We can then choose the remaining vertex from the 9 points that are not A, B, or C. This gives us a total of 9 possible choices for D.

Therefore, the number of triangles that contain A as a vertex is 1 * 9 = 9.

Similarly, we can count the number of triangles that contain B, C, D, E, F, G, H, I, J, K, and L as vertices by considering each point in turn as one of the vertices. For example, to count the number of triangles that contain B as a vertex, we can choose two other points from the 10 remaining points (since we cannot use A or B again), which gives us a total of (10 choose 2) = 45 possible triangles. We can do this for each of the remaining points to get:

Triangles containing A: 9

Triangles containing B: 45

Triangles containing C: 45

Triangles containing D: 36

Triangles containing E: 28

Triangles containing F: 21

Triangles containing G: 15

Triangles containing H: 10

Triangles containing I: 6

Triangles containing J: 3

Triangles containing K: 1

Triangles containing L: 0

The total number of triangles is the sum of these values, which is:

9 + 45 + 45 + 36 + 28 + 21 + 15 + 10 + 6 + 3 + 1 + 0 = 229

However, we have counted each triangle three times (once for each of its vertices). Therefore, the actual number of triangles is 229/3 = 76.33, which is closest to option (c) 75.

Therefore, the answer is (c) 75.

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The value of cos(π/4) when the series is carried to three terms is 0.707107, the value of cos(π/4) when the series is carried to four terms is 0.707103 and the approximate relative error for the estimation of cos(π/4) is 0.000565%.

a) To find the value of cos(π/4) using the series expansion, we can substitute x = π/4 into the series and evaluate it to three terms:

cos(π/4) = 1 - (2!/(π/4)^2) + (4!/(π/4)^4)

Calculating each term:

2! = 2

(π/4)^2 = (3.14159/4)^2 = 0.61685

4! = 24

(π/4)^4 = (3.14159/4)^4 = 0.09663

Now, plugging the values into the series:

cos(π/4) ≈ 1 - 2(0.61685) + 24(0.09663) = 0.707107

Therefore, the value of cos(π/4) when the series is carried to three terms is approximately 0.707107.

b) To find the value of cos(π/4) using the series expansion carried to four terms, we include one more term in the calculation:

cos(π/4) ≈ 1 - 2(0.61685) + 24(0.09663) - ...

Calculating the next term:

6! = 720

(π/4)^6 = (3.14159/4)^6 = 0.01519

Now, plugging the values into the series:

cos(π/4) ≈ 1 - 2(0.61685) + 24(0.09663) - 720(0.01519) = 0.707103

Therefore, the value of cos(π/4) when the series is carried to four terms is approximately 0.707103.

c) The approximate absolute error, EA, for the estimation of cos(π/4) can be calculated by comparing the result obtained in part b with the actual value of cos(π/4), which is √2/2 ≈ 0.707107.

EA = |0.707107 - 0.707103| ≈ 0.000004

Therefore, the approximate absolute error for the estimation of cos(π/4) is approximately 0.000004.

d) The approximate relative error, εA, for the estimation can be calculated by dividing the absolute error (EA) by the actual value of cos(π/4) and multiplying by 100 to express it as a percentage.

εA = (EA / 0.707107) * 100 ≈ (0.000004 / 0.707107) * 100 ≈ 0.000565%

Therefore, the approximate relative error for the estimation of cos(π/4) is approximately 0.000565%.

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