What is an obtuse angle?

The percentage of budget that is allotted to variable expenses is 46.03%

We first have to determine the solution for what the va,riable expenses is supposed to be

$2,003 (total budget) - $1,081 (fixed expenses)

= $922

Next we will have to solve for the percentage that is the budget which is allocated to the variable expenses

This is simply written as

variable expenses / total budget * 100

($922 (variable expenses) ÷ $2,003 (total budget)) × 100 = 46.03%

Hence the percentage of budget that is allotted to variable expenses is 46.03%

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Answer:

46.03%

Step-by-step explanation:

922 ÷ 2003 x 100 which gives you 46.03

The equation for a circle with centre (h,k) and radius r is (x-h)^2 + (y-k)^2 = r^2. To find the point where the line with equation y = mx intersects the circle in the first quadrant, we can substitute y = mx into the equation for the circle.

(x-h)^2 + (mx-k)^2 = r^2
Expanding and simplifying:
x^2 - 2hx + h^2 + m^2x^2 - 2kmx + k^2 = r^2
Grouping the x terms:
(1 + m^2)x^2 - 2(h+km)x + h^2 + k^2 - r^2 = 0
This is a quadratic equation in x, which we can solve using the quadratic formula:
x = [2(h+km) ± sqrt(4(h+km)^2 - 4(1+m^2)(h^2+k^2-r^2))] / 2(1+m^2)
Simplifying:
x = (h+km) ± sqrt[(h+km)^2 - (h^2+k^2-r^2)m^2] / (1+m^2)
Since we're looking for a point in the first quadrant, we want the positive solution for x. Once we find x, we can plug it back into y = mx to get the y-coordinate of the point.
Note that there may be 0, 1, or 2 solutions depending on the specific values of h, k, r, and m. If there are 2 solutions, one will be in the first quadrant and the other in the fourth quadrant. If there is 1 solution, it will be on the border between the first and fourth quadrants. If there are 0 solutions, the line does not intersect the circle in the first quadrant.

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The direction of the resultant vector is 251.57°

From the graph, we see that we have two vectors w = (10, 4) and v = (-14 , -16). Re-writing both vectors in component form, we have that

W = 10i + 4j and

v = -14i - 16j

So, the resultant vector is the sum of both vectors.

So, we have that

R = w + v

= 10i + 4j + (-14i - 16j)

= 10i - 14i + 4j - 16j

= -4i - 12j

So, the direction of the resultant vector is given by Ф = tan⁻¹(y/x) where y = -12 and x = -4

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

Ф = tan⁻¹(y/x)

Ф = tan⁻¹(-12/-4)

Ф = tan⁻¹(3)

= 71.57°

Since the vector is in the 3rd quadrant, its directions is Ф = 180° + 71.57° = 251.57°

So, the direction is 251.57°

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The key words in the question are "descriptive statistics." "Descriptive" refers to describing or summarizing data, while "statistics" refers to the collection, analysis, and interpretation of data.

This question relates to the topic of statistics.

To solve this question, you need to identify which situation involves the use of descriptive statistics. You can do this by understanding that descriptive statistics involves summarizing or describing data, such as calculating measures of central tendency (like the mean or median) or analyzing the distribution of data.

The correct answer choice is C) A study shows that the average student leaves a four-year college with a student loan debt of more than $30,000. This situation involves the use of descriptive statistics because it describes the average amount of student loan debt for a particular group of people (students who attend four-year colleges).

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Answer:

62%

Step-by-step explanation:

Its addition, 38+24=62

30+20+12=62 to make thing simpler.

Answer:

~14.4

Step-by-step explanation:

You can use the Pythagorean Theorem to find the hypotenuse (corner to corner) of the line between the corners of the rectangle.

A^2 + b^2 = c^2

8^2 + 12^2 = 208
√208 = 14.42

Answer:

The formula for the surface area, S, of a cube with edges of length 2x is:

S = 6(2x)^2

Simplifying the expression inside the parentheses gives:

S = 6(4x^2)

Multiplying 6 by 4x^2 gives:

S = 24x^2

Therefore, the formula for the surface area of a cube with edges of length 2x is S = 24x^2, which is option (B).

Step-by-step explanation:

The required expression is v = √2ay.

Let acceleration is given by: a

and initial velocity is given by: v

So, The maximum height(y) is

y= v² sinθ / 2a

As, θ=90 then

y = v² /2a

v² = 2ay

v = √2ay

Thus, the required expression is v = √2ay.

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Max had picked 24 apples.

We have,

Let x be the number of apples Max picked.

According to the problem, if he had picked twice as many apples, he would have 24 more apples than he currently has.

This can be expressed as:

2x = x + 24

Simplifying and solving for x:

x = 24

Therefore,

Max had picked 24 apples.

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(a) The value of the expression -(a%b-c/d+e*f) is -27.5 when a = 10, b = 3, c = 5, d = 2, e = 4, and f = 6

(b) For the second expression the final result is : FALSE

a. -(a%b-c/d+e*f)

Step 1: Let's assume that a = 10, b = 3, c = 5, d = 2, e = 4, and f = 6.
Step 2: Evaluate the expression inside the parentheses: c/d = 5/2 = 2.5
Step 3: Evaluate the expression inside the parentheses: e*f = 4*6 = 24
Step 4: Evaluate the expression inside the parentheses: a%b = 10%3 = 1
Step 5: Add the results of steps 2, 3, and 4: 2.5 + 24 + 1 = 27.5
Step 6: Negate the result of step 5: -27.5

Therefore, the value of -(a%b-c/d+e*f) is -27.5 when a = 10, b = 3, c = 5, d = 2, e = 4, and f = 6.

b. ! ((a>b) && (cb) = false, (cb) && (c b) && (c > d))
  Step 1: a > b = 6 > 4 = true
  Step 2: c > d = 8 > 2 = true
  Step 3: true && true = true
  Step 4: !(true) = false
  Final result: false

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The correct answer is:
a. V1, V2 form the basis; V3 = 4V1 + V2, V4 = -4V1 + V2

To find a subset of the vectors that forms a basis for the space spanned by the vectors and express each vector that is not in the basis as a linear combination of the basis vectors, follow these steps:

1. Write the given vectors as rows of a matrix:
  A = | 1   0  1  1 |
      |-7   7 -4  1 |
      |-3   7  0  5 |
      |-11  7 -8 -3 |

2. Perform Gaussian elimination to find the row-reduced echelon form (RREF) of the matrix A.

3. The RREF of matrix A is:
  RREF(A) = | 1  0  1  1 |
            | 0  1 -2  3 |
            | 0  0  0  0 |
            | 0  0  0  0 |

4. Identify the pivot columns in the RREF matrix. In this case, the first and second columns have pivots.

5. The pivot columns correspond to the original vectors that form a basis. In this case, V1 and V2 form the basis.

6. Express each vector that is not in the basis as a linear combination of the basis vectors. For V3 and V4, we can see that:
  V3 = 4V1 + V2
  V4 = -4V1 + V2

So, the correct answer is:
a. V1, V2 form the basis; V3 = 4V1 + V2, V4 = -4V1 + V2

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Answer: For the first one

9037 and 21800

Step-by-step explanation:

Add them all up.

Answer:

225 cm²

Step-by-step explanation:

The container has 2.7L of water in it but it is only 4/5 full

Therefore if the container were to be filled entirely with water  it would contain
2.7 x 5/4 = 3.375 Liters

This, therefore is the volume of the container is 3.375 L
3.375 L = 3.375 x 1000 cm³
             =  3, 375 cm³

The volume of a cube of side a is given by
V = a³

The base area of a cube of side a is given by
A = a²

We have calculated the volume of the cube as 3.375 cm³

Therefore each side of the cubical container
[tex]a = \sqrt[3]{3375} = 15[/tex] cm

The base area is
a² = 15²
    = 225 cm²

According to the given information :

(1) 2% of the calories in one serving of Fattoush come from carbohydrates.

(2) If the man consumes one serving of Fattoush, which contains 21.39g of protein, he still needs to consume an additional 46.45g of protein during the rest of the day.

1. To calculate the percent of Fattoush calories that comes from fat and carbohydrates, we need to know the total number of calories in one serving of Fattoush. Let's assume that the total number of calories in one serving of Fattoush is 200.

To calculate the percent of calories from fat:
- We know that 1g of fat = 9 calories, so if there are 21.39g of fat in one serving of Fattoush, we can multiply that by 9 to get the total number of calories from fat: 21.39g x 9 = 192.51 calories from fat.
- To calculate the percent of calories from fat, we can divide the total calories from fat (192.51) by the total number of calories in one serving of Fattoush (200) and then multiply by 100: (192.51 / 200) x 100 = 96.3%.

So, 96.3% of the calories in one serving of Fattoush come from fat.

To calculate the percent of calories from carbohydrates:
- We know that 1g of carbohydrates = 4 calories, so if there are 1g of carbohydrates in one serving of Fattoush, we can multiply that by 4 to get the total number of calories from carbohydrates: 1g x 4 = 4 calories from carbohydrates.
- To calculate the percent of calories from carbohydrates, we can divide the total calories from carbohydrates (4) by the total number of calories in one serving of Fattoush (200) and then multiply by 100: (4 / 200) x 100 = 2%.

So, 2% of the calories in one serving of Fattoush come from carbohydrates.

2. To calculate the RDA for protein for a 187# man, we need to convert his weight from pounds to kilograms:
- 187# / 2.205 = 84.8 kg
- RDA = 0.8g protein per kg of body weight
- RDA = 0.8 x 84.8 = 67.84g of protein

If the man consumes one serving of Fattoush, which contains 21.39g of protein, he still needs to consume an additional:
- 67.84g - 21.39g = 46.45g of protein during the rest of the day.

3. If we add 3 oz of grilled chicken to one serving of Fattoush, the new total calories would be:
- 200 (calories in one serving of Fattoush) + 130 (calories from 3 oz of grilled chicken) = 330 total calories

To calculate the new percentages of calories from fat and carbohydrates:
- We know that 1g of fat = 9 calories, so if there are now 24.39g of fat in the dish (21.39g from the Fattoush and 3g from the chicken), we can multiply that by 9 to get the total number of calories from fat: 24.39g x 9 = 219.51 calories from fat.
- To calculate the percent of calories from fat, we can divide the total calories from fat (219.51) by the total number of calories in the dish (330) and then multiply by 100: (219.51 / 330) x 100 = 66.5%.

So, the percent of calories from fat has decreased from 96.3% to 66.5%.

- We know that there is 1g of carbohydrates in one serving of Fattoush and 1g of carbohydrates in the chicken, so the total number of calories from carbohydrates is: 1g x 4 = 4 calories from carbohydrates.
- To calculate the percent of calories from carbohydrates, we can divide the total calories from carbohydrates (4) by the total number of calories in the dish (330) and then multiply by 100: (4 / 330) x 100 = 1.2%.

So, the percent of calories from carbohydrates has slightly decreased from 2% to 1.2%.

The percent increase in total calories is:
- We know that the original total number of calories in one serving of Fattoush was 200.
- We added 130 calories from the chicken.
- The percent increase in total calories is (130 / 200) x 100 = 65%.

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Bruce Lovegren paid a monthly annual premium of $125 for his $15,000 10-year term life insurance policy.

The monthly premium can be calculated by dividing the total cost of the policy by the number of months in the policy term:

monthly premium = total cost of policy / number of months in policy term

The total cost of the policy can be calculated by multiplying the annual premium by the number of years in the policy term:

total cost of policy = annual premium * number of years in policy term

The number of months in a year is 12.

Bruce Lovegren purchased the policy on April 14, 1977, so the policy was in effect for 10 years and 8 months, or 128 months.

The annual premium as follows:

total cost of policy = $15,000

number of years in policy term = 10

annual premium = total cost of policy / number of years in policy term

annual premium = $15,000 / 10

annual premium = $1,500

monthly premium = annual premium / 12

monthly premium = $1,500 / 12

monthly premium = $125

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You might need to transfer measurements from the metric system to the US system and vice versa in various situations.

Some examples include:

1. International trade: If you are exporting or importing goods, it's essential to convert measurements to the recipient's preferred system, ensuring proper understanding of product specifications.

2. Travel: When traveling to a different country, you may need to convert distances, speed limits, or temperatures to better understand local road signs or weather conditions.

3. Cooking and recipes: When following a recipe from a different country, converting ingredient measurements can be crucial for accurate results.

4. Construction and engineering: Working on projects with international collaboration may require converting measurements to ensure all parties understand the specifications.

5. Science and education: As the metric system is the standard for scientific research, it may be necessary to convert US measurements to metric for consistency in data reporting and understanding.

Remember to use appropriate conversion factors when transferring measurements between the metric system and the US system to ensure accurate results.

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≈≈≈Answer:

Step-by-step explanation:

To calculate the total amount Karl will pay to the bank, we need to find the monthly payment and then multiply it by the total number of payments over three years.

First, we need to calculate the monthly interest rate. Since the loan is compounded monthly, we divide the annual interest rate by 12:

Monthly interest rate = 7.25% / 12 = 0.006041667

Next, we need to calculate the total number of payments Karl will make over the course of the loan. Since he will be making monthly payments for three years, there will be a total of:

Total number of payments = 3 years x 12 months/year = 36 payments

To calculate the monthly payment, we can use the formula for the present value of an annuity:

Monthly payment = P * (r / (1 - (1 + r)^(-n)))

where P is the principal amount (in this case, $5000), r is the monthly interest rate, and n is the total number of payments.

Plugging in the values, we get:

Monthly payment = [tex]5000 * \frac{0.006041667}{(1-(1+0.006041667^{-36} )} = $154.96[/tex]

Finally, we can calculate the total amount Karl will pay to the bank by multiplying the monthly payment by the total number of payments:

Total amount paid = Monthly payment x Total number of payments = $154.96 x 36 = $5,578.56

Therefore, the total amount Karl will pay to the bank is $5,578.56

Option c) has moderate variability and is closer to the true proportion and thus is the most likely sequence of sample proportions to occur for 5 random samples of 8 students from this population.

To determine which sequence of sample proportions is most likely to occur for 5 random samples of 8 students from a population where 60% are eligible for financial aid, we'll examine the variability in the sample proportions generated with the applet.

Given the options:
a) 0.250, 0.125, 0.500, 0.500, 0.875
b) 0.600, 0.600, 0.600, 0.600, 0.600
c) 0.375, 0.625, 0.500, 0.500, 0.875

Option b) has no variability, which is unlikely in random sampling.

Option a) has very high variability, which is also unlikely.

Option c) has moderate variability and is closer to the true proportion of 0.60, making it the most likely sequence of sample proportions to occur for 5 random samples of 8 students from this population.

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Answer:

Step-by-step explanation:To determine the line of reflection, we need to find the equation of the line that is equidistant from each vertex of the original triangle and the corresponding vertex of the reflected triangle.

First, let's find the coordinates of the image of each vertex under the reflection. Since F' is given as (2, 2), we can reflect F across the unknown line of reflection to find the image of D and E. The line of reflection must be equidistant from each of these pairs of corresponding points.

To reflect F across a vertical line, the x-coordinate of F' must be the same as that of F but with the opposite sign. The x-coordinate of F is -2, so the x-coordinate of its image F' must be 2. Similarly, the y-coordinate of F' is 2, which means that the line of reflection must pass through the point (2, 2).

To reflect D across the same line, we can draw a perpendicular bisector between D and its image D', which must intersect the line of reflection at a right angle. The midpoint of DD' lies on the line of reflection, and it is equidistant from D and D'. Using the midpoint formula, we find the midpoint of DD' to be ((-3+2)/2, (5+2)/2) = (-0.5, 3.5). Since this point lies on the line of reflection, we can use the point-slope form of a line to find the equation of the line passing through (2, 2) and (-0.5, 3.5):

(y - 2) = m(x - 2) (where m is the slope of the line of reflection)

Simplifying:

y - 2 = m(x - 2)

y = mx - 2m + 2

To find the value of m, we can use the fact that the midpoint of DE lies on the line of reflection as well. The midpoint of DE is ((-3-10)/2, (5+4)/2) = (-6.5, 4.5). Substituting these values into the equation of the line, we get:

4.5 = m(-6.5) - 2m + 2

2.5 = -8.5m

m = -0.294

Therefore, the equation of the line of reflection is:

y = -0.294x + 2.588

This line is not the x-axis, y-axis or the line y=x. Therefore, the line of reflection is neither the x-axis nor the y-axis, and it is not the line y = x.

Answer:

X-axis

Step-by-step explanation:

I am in the middle of taking the quiz and this is the answer I think would be correct!

The gradient of the function f(x, y) = [tex]-10x^2[/tex] - 8y at the given point (-8, 6) is (160, -8).

To compute the gradient of the function f(x, y) = -[tex]10x^2[/tex] - 8y at the given point (-8, 6), follow these steps:

1. Find the partial derivatives of f with respect to x and y.

2. Evaluate the partial derivatives at the given point.

3. Combine the partial derivatives into a gradient vector.

Step 1: Find the partial derivatives.

∂f/∂x = -20x

∂f/∂y = -8

Step 2: Evaluate the partial derivatives at the given point (-8, 6).

∂f/∂x at (-8, 6) = -20(-8) = 160

∂f/∂y at (-8, 6) = -8

Step 3: Combine the partial derivatives into a gradient vector.

Gradient = (∂f/∂x, ∂f/∂y) = (160, -8)

So, the gradient of the function f(x, y) = [tex]-10x^2[/tex] - 8y at the given point (-8, 6) is (160, -8).

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The simplified frame of the expression [tex]3 \times \frac{7 {}^{2} }{3} \times \frac{8}{3} \times 7- \frac{1}{4} [/tex] is 6511.5.

To disentangle the given expression, we got to apply the arrange of operations (PEMDAS) and streamline the terms utilizing the example and division rules.

PEMDAS stands for Enclosures, Exponents, Multiplication and Division, and Expansion and Subtraction. We ought to perform the operations in this arrange to streamline the expression.

We rearrange the type: [tex]7^2 = 49[/tex]

We rearrange the division 8/3 by partitioning the numerator by the denominator: [tex] \frac{8}{3} = 2 \frac{2}{3} [/tex]

We disentangle the division 7-1/4 utilizing the run the show that a negative example is comparable to the corresponding of the base raised to the positive type: 7-1/4 = 1/74.

To revamp the expression with the rearranged values:  [tex]3 \times 49 / (2 \frac{2}{3} \times 1/74)[/tex]

To partition divisions, we increase by the corresponding of the second division:  [tex]3 \times 49 \times 74 / (2 \frac{2}{3} )[/tex]

We have to be rearrange the blended number  [tex]2 \frac{2}{3} [/tex] by increasing the total number by the denominator and including the numerator: [tex]2 \frac{2}{3} = \frac{8}{3} [/tex]

We will disentangle the expression by canceling out common variables:

3 x 49 x 74 / (8/3) = 6511.5

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a) A nonzero element X in W is:

X = [ -10  -4 ]
       [   8    6 ]

b) The matrix Y is in V because it's a symmetric 2x2 matrix, but it's not in W since it can't be formed by any linear combination of matrix A.

a. To find a nonzero element X in W, we need to find a linear combination of the given matrix in the span of W. Let's denote the given matrix as A:

A = [ -5  -2 ]
       [  4    3 ]

Since W = span{A}, a linear combination of A would be:

X = k * A

where k is any scalar value. Let's choose k = 2:

X = 2 * A = [ -10  -4 ]
                [   8    6 ]

So, a nonzero element X in W is:

X = [ -10  -4 ]
       [   8    6 ]

b. To find an element Y in V (the vector space of symmetric 2x2 matrices) that is not in W, we need a matrix that cannot be formed by any linear combination of the given matrix A.

A symmetric 2x2 matrix has the form:

Y = [ a  b ]
       [ b  c ]

Let's choose a symmetric matrix that doesn't have the same pattern as A. For example:

Y = [ 1  2 ]
       [ 2  1 ]

This matrix Y is in V because it's a symmetric 2x2 matrix, but it's not in W since it can't be formed by any linear combination of matrix A.

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The maximum profit occurs when 34 offices are painted per month, and the maximum profit is R13500.

Based on the given information, Brush Pro is an one-man paint business owned by Banele and their monthly profit, P, depends on the number of offices painted per month, x. The profit function is given as P(x) = -x + 27x^2 + 132x + 2970, where 0 < x < 34.

To find the maximum profit, we need to find the vertex of the parabolic function. The vertex is located at x = -b/2a, where a = 27, b = 132.

x = -132/(2*27) = -2.44

Since x must be between 0 and 34, the maximum profit will occur at x = 0 or x = 34. We need to evaluate P(0) and P(34) to determine which one is the maximum.

P(0) = -0 + 27(0)^2 + 132(0) + 2970 = 2970
P(34) = -34 + 27(34)^2 + 132(34) + 2970 = 13500

Therefore, the maximum profit occurs when 34 offices are painted per month, and the maximum profit is R13500.

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The expression to show how much Mr. Levy will earn in h hours is A = 8h.

In 6 hours, Levy can earn $48.

We have,

From the table,

4 hours = $32

7 hours = $56

This means,

1 hour = $8

Now,

We can have ordered pairs as:

(1, 8), (4, 32), and (7, 56)

The expression for the amount earned in h hours.

A = mh + c

m = (32 - 8)/(4 - 1) = 24/3 = 8

(1, 8) = (h, A)

8 = 8 x 1 + c

8 = 8 + c

c = 8 - 8

c = 0

Now,

The expression is A = 8h

Now,

For A = 48

48 = 8h

h = 6

This means,

In 6 hours, Levy can earn $48.

Thus,

The expression to show how much Mr. Levy will earn in h hours is A = 8h.

In 6 hours, Levy can earn $48.

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Larry can have 5 tanks of goldfish and 3 tanks of minnows, with 5 goldfish and 5 minnows in each tank.

To find out how many goldfish and how many minnows will be in each tank, we need to find the greatest common divisor (GCD) of 25 and 15, which represents the largest number of fish that can be evenly divided into both groups.

The prime factorization of 25 is 55, and the prime factorization of 15 is 35, so the GCD of 25 and 15 is 5.

This means that Larry can put 5 goldfish and 5 minnows in each tank, and he will have:

25 / 5 = 5 tanks of goldfish

15 / 5 = 3 tanks of minnows

So Larry can have 5 tanks of goldfish and 3 tanks of minnows, with 5 goldfish and 5 minnows in each tank.

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Answer:

I assume that "Rud" is a typo and you mean "Sin" instead.

To find the derivative of Sin(x^2) - Cos(x), we need to use the chain rule and the derivative of the trigonometric functions.

The derivative of Sin(x^2) is:

d/dx [Sin(x^2)] = Cos(x^2) * d/dx [x^2] = 2x * Cos(x^2)

The derivative of -Cos(x) is:

d/dx [-Cos(x)] = Sin(x)

Therefore, the derivative of the function Sin(x^2) - Cos(x) is:

2x * Cos(x^2) + Sin(x)

So the answer is option (b) -2xcos(x^2) + sin(x).

The answer is option (b): -2xcos(x^2).

Assuming that "Rud-cost" is a typo and the function is meant to be "Rudin-cost", which is a function defined as:

Rudin-cost(x) = cos(x^2)

To find the derivative of Rudin-cost(x), we can use the chain rule and the power rule for differentiation. Specifically, if we let u = x^2, then we have:

Rudin-cost(x) = cos(u)

Using the chain rule, we get:

Rudin-cost'(x) = -sin(u) * u'

where u' is the derivative of u with respect to x, which is:

u' = d/dx(x^2) = 2x

Substituting this back into the expression for Rudin-cost'(x), we get:

Rudin-cost'(x) = -sin(x^2) * 2x

Therefore, the answer is option (b): -2xcos(x^2).

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The tables are completed as follows:

For Table 1, we have an exponential function defined as follows:

f(x) = b^x.

We have that when x = 0.699, f(x) = 5, hence the base b is obtained as follows:

b^0.699 = 5

b = 5^(1/0.699)

b = 10.

Hence, when x = 1.477, the value of f(x) is given as follows:

f(x) = 10^1.477

f(x) = 30.

Table 2 is the inverse of table 1. For Table I, we have that when x = 1.301, f(x) = 20, hence for table 2, we have that f(x) = 1.301 when x = 20.

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1/5

you take 1/4÷ 1 1/4 and you get 1/5

The given function is a valid probability density function.

We have,

B.

As f(x) ≥ 0 for all values of x and the total area under the density function above the x-axis is 1, the given function is a valid probability density function.

(a)

P(x < 6) = 0.5 (area of the rectangle with base 6 and height 0.1)

(b) P(x > 5) = 0.3 (area of the triangle with base 1 and height 0.3)

(c) P(4 < x < 8) = 0.8 (area of the rectangle with base 4 and height 0.1 plus the area of the triangle with base 4 and height 0.7 plus the area of the rectangle with base 2 and height 0.1)

(d) P(6 < x < 7) = 0.4 (area of the rectangle with base 1 and height 0.4)

Thus,

The given function is a valid probability density function.

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a)  the risk-neutral partner wishes to initiate the project, but the risk-averse partner uses their veto power to block it.

b) with the proposed compensation of 50, both partners would accept the proposal and initiate the project.

a. We can calculate the expected payoff of the project as follows:

E(Payoff) = πH - C(1-π)

Substituting the given values, we get:

E(Payoff) = (1/2)(2000) - 900

E(Payoff) = 100

The risk-neutral partner would initiate the project because the expected payoff is positive. However, the risk-averse partner would use their veto power to block the project because they are risk-averse and the expected payoff is not guaranteed.

b. Let's calculate the expected payoff for each partner with the proposed compensation:

For the risk-neutral partner, the expected payoff becomes:

E(Payoff1) = π(H - 50) - C(1 - π)

Substituting the given values, we get:

E(Payoff1) = (1/2)(1950) - 900

E(Payoff1) = 75

For the risk-averse partner, the expected payoff becomes:

E(Payoff2) = πH - C(1 - π) + 50(1 - π)

Substituting the given values, we get:

E(Payoff2) = (1/2)(2000) - 900 + 50(1/2)

E(Payoff2) = 125

Both partners would accept the proposal and initiate the project because the expected payoff for each partner is positive. The risk-averse partner is willing to accept the proposal because the compensation of 50 in case of failure reduces their risk, resulting in a positive expected payoff.

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