Let z= g(x,y) = 8x^2 - y^2 + 3xy. Find the following using the formal definition of the partial derivativea. ϑz/ϑxb. ϑz/ϑyc. ϑg/ϑy(-1,2)d. g_x(-1,2)

The width of the sidewalk is approximately 3.21 feet (rounded to two decimal places).

Let's use the given information to solve for the width (x) of the sidewalk. The area of the rectangular swimming pool is 20 feet by 30 feet, which equals 600 square feet (20*30 = 600).

The total area of the pool and sidewalk is given as 825 square feet. To find the area of just the sidewalk, we subtract the area of the pool from the total area: 825 - 600 = 225 square feet.

Now, let's express the area of the sidewalk using the dimensions of the pool and the width of the sidewalk (x). Since the sidewalk is on 3 sides, we can represent the area as follows:

2(20x) + 30x = 225

Simplify the equation:

40x + 30x = 225

Combine the terms:

70x = 225

Now, solve for x:

x = 225 / 70

x ≈ 3.21

The width of the sidewalk is approximately 3.21 feet (rounded to two decimal places).

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The values of x1 and x2 that maximize the revenue are x1 = 20 thousand units of racing bicycles and x2 = 7.5 thousand units of mountain bicycles.

To maximize the revenue, we need to find the values of x1 and x2 that maximize the function R(x1, x2) = -6x1² - 10x2² - 2x1x2 + 46x1 + 106x2.

To do this, we can take partial derivatives of R with respect to x1 and x2, set them equal to zero, and solve for x1 and x2. That is:

∂R/∂x1 = -12x1 - 2x2 + 46 = 0

∂R/∂x2 = -20x2 - 2x1 + 106 = 0

Solving these two equations simultaneously, we get:

x1 = (5/3) x2 + (23/3)

x2 = (1/5) x1 + (53/10)

Substituting the second equation into the first equation, we get:

x1 = (5/3) [(1/5) x1 + (53/10)] + (23/3)

x1 = (1/3) x1 + (89/6)

Solving for x1, we get:

x1 = 20

Substituting x1 = 20 into the second equation, we get:

x2 = (1/5) (20) + (53/10) = 7.5

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The volume of the triangular prism is 360 cubic inches

What is volume of triangular prism?

Volume = Area × Height

Here given, the base is a right triangle with a leg of 8 in. and hypotenuse of 10 in. the height of the prism is 15 in.

We want to find volume of the triangular prism.

We can find the length of the other leg of the triangle,

Height ² + Base² = Hypotenuse ²

[tex]a^2 + b^2 = c^2 \\ 8^2 + b^2 = 10^2 \\ 64 + b^2 = 100 \\ b^2 = 36 \\ b = 6[/tex]

So the base triangle is 6 in.

Area of a triangle = 1/2 × base × height

A = 1/2 × 8 in. × 6 in.

A = 24 in²

Now volume of the prism,

V = A × height

V = 24 in² × 15 in

V = 360 in³

Therefore, the volume of the triangular prism is 360 cubic inches (to the nearest tenth).

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A graph of the image of the figure after a dilation by a scale factor of 2 centered at the origin is shown below.

In Mathematics and Geometry, a dilation simply refers to a type of transformation which typically changes the size of a geometric object, but not its shape.

Next, we would apply a dilation to the coordinates of the pre-image by using a scale factor of 2 centered at the origin as follows:

Ordered pair (6, 9) → Ordered pair (6 × 2, 9 × 2) = (12, 18).

Ordered pair (6, 6) → Ordered pair (6 × 2, 6 × 2) = (12, 12).

Ordered pair (9, 9) → Ordered pair (9 × 2, 9 × 2) = (18, 18).

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Missing information:

The question is incomplete and the complete question is shown in the attached picture.

The following mathematical operation must be carried out in order to determine a cylinder's volume: V = h r².

How can I calculate a cylinder's volume?

We must work out the following mathematical equation in order to determine a cylinder's volume:

V = Πhr² (h = height of cylinder, r= radius)

Let's use an instance.

The size of our example cylinder is 6 centimetres in diameter and 10 centimetres height. What is the size of it?

We substitute the values as follows to determine the volume:

Volume = 3.1415 x 10 cm x 3 cm²

= 307.35 cm³

Therefore, The following mathematical operation must be carried out in order to determine a cylinder's volume: V = h r².

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False.

Mutually exclusive means that the occurrence of event A and the occurrence of event B cannot happen at the same time.

In this case, the occurrence of event A does affect the probability of the occurrence of event B because if event A occurs, then event B cannot occur.

Independent means that the occurrence of event A has no effect on the probability of the occurrence of event B. In this case, the occurrence of event A does not prevent the occurrence of event B.

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To find the slope of the tangent to the curve r = -6 - 2 cos θ at the value θ = x/2, we first need to find the rectangular coordinates (x, y) using the polar coordinates (r, θ). The rectangular coordinates can be found using the following equations:
x = r * cos(θ)
y = r * sin(θ)

Next, we need to differentiate both x and y with respect to θ:
dx/dθ = dr/dθ * cos(θ) - r * sin(θ)
dy/dθ = dr/dθ * sin(θ) + r * cos(θ)
Now, we find the derivative of r with respect to θ:
r = -6 - 2 cos(θ)
dr/dθ = 2 sin(θ)
Then, we plug in θ = x/2 and evaluate x and y:
x = r * cos(x/2)
y = r * sin(x/2)
Now, we evaluate dx/dθ and dy/dθ at θ = x/2:
dx/dθ = 2 sin(x/2) * cos(x/2) - r * sin(x/2)
dy/dθ = 2 sin(x/2) * sin(x/2) + r * cos(x/2)
Finally, the slope of the tangent (m) is given by:
m = dy/dθ / dx/dθ
Plug in the values of dy/dθ and dx/dθ that we've calculated and simplify to find the slope of the tangent at the given point.

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Answer math suck a

Step-by-step explanation:

The probability of drawing a red card is always exactly 1/2

We can approach this problem using conditional probability. Let A denote the event that the first card is red, and B denote the event that the second card is red. Then, using the law of total probability, we have:

P(B) = P(A)P(B|A) + P(A^c)P(B|A^c)

where P(A) = 1/2, P(A^c) = 1/2, P(B|A) = 25/51 (since there are 25 red cards left out of 51 total cards), and P(B|A^c) = 26/51 (since there are 26 red cards left out of 51 total cards).

Therefore, we have:

P(B) = (1/2)(25/51) + (1/2)(26/51) = 51/102 = 1/2

This means that the probability of drawing two red cards in a row is exactly 1/2, regardless of the order in which the cards are drawn.

Similarly, we can calculate the probability of drawing three red cards in a row as follows:

P(C) = P(A)P(B|A)P(C|AB) + P(A)P(B^c|A)P(C|A(B^c)) + P(A^c)P(B|A^c)P(C|A^cB) + P(A^c)P(B^c|A^c)P(C|A^cB^c)

where C denotes the event that the third card is red, and we have conditioned on the first two cards that were drawn. Using the same reasoning as before, we have:

P(C) = (1/2)(25/51)(24/50) + (1/2)(26/51)(25/50) + (1/2)(26/51)(25/50) + (1/2)(25/51)(24/50) = 1225/5100 = 49/204

Thus, the probability of drawing three red cards in a row is less than 1/2, and in general, the probability of drawing n red cards in a row is (1/2)^n. Therefore, there is no strategy that can give you a better than 1/2 chance of winning the game, as the outcome of each draw is independent and the probability of drawing a red card is always exactly 1/2

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The estimated expectation of the given observations is 61.00 meters, and the estimated standard deviation is 1.27 meters.

These estimates are obtained using the sample mean and sample standard deviation formulae, which are unbiased estimators of the population mean and population standard deviation, respectively.

To estimate the population mean, we calculate the sample mean as the sum of the observations divided by the sample size, which is 61.00 meters. To estimate the population standard deviation, we calculate the sample standard deviation as the square root of the sum of the squared deviations of each observation from the sample mean divided by the sample size minus one, which is 1.27 meters.

The given information, Στ = -550.28 and Σα? = 33656.86, can be used to check the accuracy of the estimates.

The sum of the squared deviations of each observation from the sample mean multiplied by the sample size minus one is equal to the sum of squares of deviations from the population mean multiplied by the sample size minus one, which is denoted as Σ(Xi - μ)2 = (n-1)σ2. Using these formulae, we can calculate the sample mean and sample standard deviation and verify the given information.

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The derivative of the function g(x) = (5x2 + 4x - 4)e is g'(x) = (10x + 4)e + (5x2 + 4x - 4)(e).

To find the derivative of the function g(x) = (5x2 + 4x - 4)e, we can use the product rule of differentiation. The product rule states that if we have two functions u(x) and v(x), then the derivative of their product is given by:

(uv)' = u'v + uv'

In this case, we can take u(x) = 5x2 + 4x - 4 and v(x) = e. Then, using the power rule and the fact that the derivative of e to any power is e to the same power, we get:

u'(x) = 10x + 4

v'(x) = e

Putting it all together, we get:

g'(x) = (5x2 + 4x - 4)'e + (5x2 + 4x - 4)(e)'

g'(x) = (10x + 4)e + (5x2 + 4x - 4)(e)

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Picking the white ball twice (T) is 1/4

Explanation: In this problem, we have a box with two balls - one white and one red. We will draw a ball from the box, put it back, and then draw a second ball (sampling with replacement). Let T be the event of getting the white ball twice, F the event of picking the white ball first, and S the event of picking the white ball in the second drawing.
To compute P(T), we need to find the probability of picking the white ball in both drawings:
P(T) = P(F) * P(S|F)
Since there is one white ball and one red ball in the box, the probability of picking the white ball first (F) is 1/2. Since we're sampling with replacement, the probability of picking the white ball in the second drawing (S) given that the white ball was picked first (F) is also 1/2.
So, the probability of picking the white ball twice (T) is:
P(T) = (1/2) * (1/2) = 1/4
Therefore, the probability of picking the white ball twice (T) is 1/4.

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There are 13 students in the group who are not enrolled in either a

mathematics course or a physics course.

We can solve this problem using the principle of inclusion-exclusion,

which states that the size of the union of two sets is given by:

|A ∪ B| = |A| + |B| - |A ∩ B|

where |A| represents the size (number of elements) of set A, and |A ∩ B|

represents the size of the intersection of sets A and B.

In this case, we want to find the number of students who are not enrolled

either a mathematics course or a physics course.

Let M be the set of students enrolled in a mathematics course, and let P

the set of students enrolled in a physics course. Then the number of

students who are not enrolled in either course is:

|not enrolled| = |total| - |M ∪ P|

We are given that |M| = 15, |P| = 10, and |M ∩ P| = 5. To find |M ∪ P|, we

use the inclusion-exclusion principle:

|M ∪ P| = |M| + |P| - |M ∩ P|

= 15 + 10 - 5

= 20

So the number of students who are not enrolled in either course is:

|not enrolled| = |total| - |M ∪ P|

= 33 - 20

= 13

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False. The probability of independent events each occurring is P(A) x P(B), not P(A) + P(B).

The possibility of event A and event B occurring collectively may be calculated using the multiplication rule of chance, which states that the probability of two independent events taking place collectively is same to the made of their person chances.

Consequently, the chance of A and B each taking place together may be calculated as P(A) x P(B), assuming that a and B are independent activities.

it's far critical to notice that the addition rule of possibility can best be carried out when events A and B are jointly unique, which means they can not arise together. in that case, the chance of both event A or event B taking place may be calculated by using including their character possibilities.

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The company should hire students for 10 hours of labor per day to maximize profit.

To determine the number of hours (L) of labor the company should use per day to maximize profit, we need to consider the revenue, cost, and profit functions, and then find the critical points.

1. Revenue: The company sells 3L^(2/3) Kg of wild mushrooms at $15.00 per Kg.

So, the revenue function R(L) = 15 * 3L^(2/3) = 45L^(2/3).

2. Cost: The company pays $6.00 per hour for L hours of labor.

So, the cost function C(L) = 6L.

3. Profit: The profit function P(L) is the difference between revenue and cost:

P(L) = R(L) - C(L) = 45L^(2/3) - 6L.

4. To maximize profit, we need to find the critical points by taking the derivative of the profit function and setting it equal to zero:

P'(L) = d(45L^(2/3) - 6L) / dL = 0.

5. Derivative:

P'(L) = (2/3)*45L^(-1/3) - 6.

6. Solve for L:

Set P'(L) = 0 and solve for L:

(2/3)*45L^(-1/3) - 6 = 0.

By solving this equation, we find that L ≈ 9.58 hours.

Since the company cannot hire a fraction of an hour, it should hire students for 10 hours of labor per day to maximize profit.

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The 99% confidence interval for the mean score of all students is constructed as (85.497, 96.5026).

Given,

Sample size, n = 30

Sample mean, x = 91

Standard deviation, s = 11.7

The 99% confidence interval for the mean score of all students can be calculated as,

x ± z [tex]\frac{s}{\sqrt{n} }[/tex]

z value for 99% confidence interval = 2.576

Confidence interval = (91 ± (2.576 × 11.7/√30)

                                 = (91 ± 5.5026)

                                 = (85.497, 96.5026)

Hence the confidence interval is (85.497, 96.5026).

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Answer:

Step-by-step explanation:

Choco bar cost you $0.75 ea, selling price $2.00, profit for each sale $1.25

Ice cream sandwich cost you $0.85 each, selling price $2.25, profit $1.40

Frozen fruit bar cost you $0.50 each, selling price $1.80, profit $1.30

Step-by-step explanation:

Let's denote the number of Choco bars as "c", the number of ice-cream sandwiches as "I", and the number of frozen fruit bars as "F".

To find the inequality to represent the number of each item to make a profit of at least $700, we need to use the information given in the problem.

The profit from selling one Choco bar is $1.25, the profit from selling one ice cream sandwich is $1.40, and the profit from selling one frozen fruit bar is $1.30.

The total profit can be calculated by multiplying the profit per item with the number of items sold and adding the profits from each item. Therefore, we can write:

Total Profit ≥ $700

1.25c + 1.40I + 1.30F ≥ 700

This is the inequality that represents the number of chocolate fudge bars, ice-cream sandwiches, and frozen fruit bars that will make a profit of at least $700.

There are 42 subsets of size two that contain at least one of the elements of {1, 2, 3}.

There are [tex]${8\choose2}=28$[/tex] subsets of size two that can be formed from the set {1, 2, 3, 4, 5, 6, 7, 8}.

To count the number of subsets of size two that contain at least one of the elements of {1, 2, 3}, we can use the principle of inclusion-exclusion.

Let A be the set of subsets of size two that contain 1, B be the set of subsets of size two that contain 2, and C be the set of subsets of size two that contain 3. We want to count the size of the union of these three sets, i.e., the number of subsets of size two that contain at least one of the elements of {1, 2, 3}.

By the principle of inclusion-exclusion, we have:

|A ∪ B ∪ C| = |A| + |B| + |C| - |A ∩ B| - |A ∩ C| - |B ∩ C| + |A ∩ B ∩ C|

To calculate the sizes of these sets, we can use combinations. For example, |A| is the number of subsets of size two that can be formed from {1, 2, 3, 4, 5, 6, 7, 8} with 1 as one of the elements. This is equal to [tex]${3\choose1}{5\choose1}=15$[/tex], since we must choose one of the three elements in {1, 2, 3} and one of the five remaining elements.

Similarly, we have:

|A| = [tex]${3\choose1}{5\choose1}=15$[/tex]

|B| = [tex]${3\choose1}{5\choose1}=15$[/tex]

|C| = [tex]${3\choose1}{5\choose1}=15$[/tex]

|A ∩ B| = [tex]${2\choose1}{5\choose0}=2$[/tex], since there are two elements in {1, 2} that must be included in the subset, and we can choose the other element from the remaining five.

|A ∩ C| = [tex]${2\choose1}{5\choose0}=2$[/tex]

|B ∩ C| = [tex]${2\choose1}{5\choose0}=2$[/tex]

|A ∩ B ∩ C| = [tex]${3\choose2}=3$[/tex], since there are three elements in {1, 2, 3} and we must choose two of them.

Substituting these values into the inclusion-exclusion formula, we get:

|A ∪ B ∪ C|[tex]= 15 + 15 + 15 - 2 - 2 - 2 + 3 = 42[/tex]

Therefore, there are 42 subsets of size two that contain at least one of the elements of {1, 2, 3}.

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The value of y for the function y = cos(-60°) is y = -1/2, option A is correct.

An equation with trigonometric functions that holds true for all of the variables in it is known as a trigonometric identity. A few normal geometrical characters incorporate the Pythagorean personality, the total and distinction characters, and the twofold point characters.

Using the unit circle and the trigonometric identity for cosine, we know that:

cos(-60°) = cos(360° - 60°)

               = cos(300°)

               = cos(180° + 120°)

               = -cos(120°)

               = -1/2

Therefore, the value of y for the function y = cos(-60°) is y = -1/2.

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Answer:

Step-by-step explanation:

1-1=0

Answer:You will have 0 moms.

Step-by-step explanation:

First you take 1 away from 1.

After that you get your answer of 0.

Answer: 9+a^2 if d=10 OR 99 if d does not equal 10

The distribution of the sample mean is approximately normal with a mean of 40 and a standard deviation of 0.559.

We are required to determine the distribution of the sample mean when a random sample of size n = 16 is taken from a normal population with mean 40 and variance 5.

The distribution of the sample mean can be found using the Central Limit Theorem, which states that when a sufficiently large sample is taken from a population with any shape, the sample mean will be approximately normally distributed. In this case, we have a normal population with mean (μ) 40 and variance (σ²) 5.

1: Calculate the standard deviation (σ) from the variance:

σ = √(σ²) = √5 ≈ 2.236

2: Calculate the standard error (SE) using the sample size (n) and the population standard deviation (σ):

SE = σ/√n = 2.236/√16 = 2.236/4 = 0.559

3: Determine the distribution of the sample mean:

The sample mean will follow a normal distribution with the same mean (μ) as the population mean and a standard deviation equal to the standard error (SE).

So, the distribution of the sample mean contains a mean of 40 and a standard deviation of 0.559.

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On solving the query we can say that The trapezium is 16 cm tall as a of result.

Mathematics is concerned with numbers and their variations, equations and related structures, shapes and their places, and possible placements for them. The relationship between a collection of inputs, each of which has an associated output, is referred to as a "function". An relationship between inputs and outputs, where each input yields a single, distinct output, is called a function. Each function has a domain and a codomain, often known as a scope. The letter f is frequently used to represent functions (x). X is the input. The four main types of functions that are offered are on functions, one-to-one functions, many-to-one functions, within functions, and on functions.

The formula for a trapezoid's area is:

Area is equal to (b1+b2)*h/2.

where h is the height of the trapezium, and b1 and b2 are the lengths of the two bases.

The trapezoid's size is 48 cm2, and its bases (b1) and (b2) are each assigned lengths of 5 cm and 7 cm, respectively. In order to solve for the height (h), we may enter these values into the formula as follows:

48 = (5 + 7) * h / 2

When we simplify the equation, we obtain:

48 = 6h / 2

48 = 3h

h = 48 / 3

h = 16

The trapezium is 16 cm tall as a result.

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The antiderivative of f(x) = ³√x² + x√x is [tex]1/3 x^3/2 (2x^1/3 + 3x^1/2)[/tex] + C.

The antiderivative of a capability is the converse of the subsidiary. As such, assuming that we have a capability f(x) and we take its subordinate, we get another capability that lets us know how the first capability is changing regarding x. The antiderivative of f(x) is a capability that lets us know how the first capability changes as for x the other way. It is additionally called the endless vital of f(x).

Presently, we should view as the antiderivative of the given capability f(x) = ³√x² + x√x. We can separate it into two sections:

f(x) = ³√x² + x√x

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

To see as the antiderivative of [tex]x^(2/3)[/tex], we want to add 1 to the example and separation by the new type:

∫[tex]x^(2/3)[/tex] dx = (3/5)[tex]x^(5/3)[/tex] + C

where C is the steady of incorporation. Also, to view as the antiderivative of[tex]x^(3/2)[/tex], we add 1 to the example and separation by the new type:

∫[tex]x^(3/2)[/tex] dx = (2/5)[tex]x^(5/2)[/tex] + C

where C is the steady of incorporation.

Accordingly, the antiderivative of f(x) = ³√x² + x√x is:

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

= (3/5)[tex]x^(5/3)[/tex] + (2/5)[tex]x^(5/2)[/tex] + C

where C is the steady of incorporation.

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[tex]f(1,-2) = 1 + e^-2.[/tex]

To determine f(1,-2), we simply need to substitute 1 for x and -2 for y in the given function [tex]f(x,y) = x^2x^3+e^xy.[/tex]

[tex]f(1,-2) = 1^2 * 1^3 + e^(1*-2)[/tex]

[tex]= 1 + e^-2[/tex]

Therefore, [tex]f(1,-2) = 1 + e^-2.[/tex]

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To calculate the power of the proposed study, we need to first determine the effect size, which is the standardized difference between the mean BMI of normal and diabetic men.

The standardized difference can be calculated as:

d = (μ1 - μ2) / σ

where μ1 and μ2 are the population means of BMI for normal and diabetic men, respectively, and σ is the common population standard deviation of BMI.

From the information given in the problem, we have:

μ1 = 25

μ2 = 25 + 2.7 = 27.7

σ = 2.7

So, the effect size is:

d = (25 - 27.7) / 2.7 = -1

Next, we need to determine the significance level (α) and the sample size (n). The problem states that the sample size is 25 normal men and 25 diabetic men, so n = 50. The significance level is usually set at 0.05, which means that the probability of a Type I error (rejecting the null hypothesis when it is actually true) is 0.05.

Using a standard normal distribution table, we can find the z-score corresponding to the significance level α = 0.05:

zα = 1.645

The power of the test is the probability of correctly rejecting the null hypothesis (i.e., detecting a true difference between normal and diabetic men) when the alternative hypothesis is true. The power of a test depends on several factors, including the effect size, the significance level, the sample size, and the variability of the data.

The formula for calculating power is:

Power = P(Z > zα - d√n)

where Z is the standard normal distribution, and d and n are the effect size and sample size, respectively.

Substituting the values we have, we get:

Power = P(Z > 1.645 - (-1)√50) = P(Z > 0.843)

Using a standard normal distribution table, we can find that the probability of Z being greater than 0.843 is 0.199.

Therefore, the power of the proposed study is approximately 0.199, or 19.9%. This means that there is a 19.9% chance of correctly detecting a clinically meaningful difference in BMI between normal and diabetic men, assuming that such a difference actually exists.

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a)  the determinant of J is 1xT - 0x2 = 0, which means that the area of O(R) is 0.
b)  the determinant of J is 1x1 - 0x2 = 1, which means that the area of Q(R) is the same as the area of R, which is (13-1) x (18-6) = 144.

To find the area of O(R) using the Jacobian, we need to calculate the determinant of the Jacobian matrix of Q(u,v):

J = [∂(u+30)/∂u   ∂(u+30)/∂v  ]
    [∂(2u+Tv)/∂u  ∂(2u+Tv)/∂v]

= [1  0]
  [2  T]

(a) For R = [0,9] x [0,7], we have T = 0 since there is no v-dependence in the range of R. Therefore, the determinant of J is 1xT - 0x2 = 0, which means that the area of O(R) is 0.

(b) For R = [1,13] x [6,18], we have T = 1 since v ranges from 6 to 18. Therefore, the determinant of J is 1x1 - 0x2 = 1, which means that the area of Q(R) is the same as the area of R, which is (13-1) x (18-6) = 144.

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For a normal distribution of random variable of tread life of a particular brand, the probability a particular tire of this brand will last longer than 58,400 miles is equals to the 0.4533.

We have, tread life of a particular brand of tire is represents a random variable. It is normally distributed with mean, μ = 60,500 miles

Standard deviations, σ = 2800 miles

We have to determine probability a particular tire of this brand will last longer than 58,400 miles, P( X > 58,400). Using normal distribution the z-score formula is

[tex]z = \frac { X - \mu }{ \sigma}[/tex]

where, X --> observed value

μ --> mean

σ --> standard deviations

Here, X = 58400, substitute all known values in above formula, [tex]z = \frac { 58400- 60500}{ 2800}[/tex]

= - 0.75

Now, the required probability, P( X > 58,400 = [tex]P( \frac { X - \mu }{ \sigma} > \frac { 58400- 60500 }{ 2800})[/tex]

= P(z> -0.75)

Using the normal distribution table, the value P(z> -0.75) is equals to . So, P( X > 58400) = 0.4533. Hence, required probability value is 0.4533.

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The value of the given indefinite integrals are 4x⁴ + 3x³ + 3x² - 3x² + 3x + C and  [tex](V/2)y^{2} - 1/y + (1/3)e^{(3y) }+ C.[/tex]

Let us implement the principles to evaluate the indefinite integral, so that their values can be derived
a. integral (16x³ + 9x² + 9x² - 6x + 3)dx
= 4x⁴ + 3x³ + 3x² - 3x²+ 3x + C
here C is the constant of integration
Now let us proceed to tye next part of the question
b. integral [tex](Vy + 1/(y^{2}) + e^{(3y)}) dy[/tex]
[tex]= (V/2)y^{2} - 1/y + (1/3)e^{(3y)} + C[/tex]
here C is the constant of integration
Indefinite integral refers to a form of function which doesn't have limits to describe the family of function it belongs to.

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