Solve the equation
Complete using the provide data and solve

A) The equation relates the number of cups of flour to the number of pound cakes that the bakery makes is f = 2.75p

B) Cups of flour are needed for 14 cakes is 38.5

A) The number of cups of flour, f, that a bakery needs for the number of pound cakes that they make, p is directly proportional to each other which can be written in form ,

f/p = 8.25/3

f = (8.25/3)×p

f = 2.75 p

The equation forms is f = 2.75p

B) Cups of flour are needed for 14 cakes

Here p = 14

by putting the value in the equation we get ,

f = 2.75(14)

f = 38.5

hence , cups of flour are needed for 14 cakes is 38.5

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The area for the first shape is  16,000 square feet, the area for the second shape is 2,400 square feet. The total area of both shapes added together is 18,400 square feet.

To find the area of the first shape, which is a rectangle that is 160 feet tall and 100 feet wide, we can use the formula:

Area = length x width

So, for the first shape, the area is:

Area = 160 ft x 100 ft

Area = 16,000 square feet

To find the area of the second shape, which is a rectangle that is 60 feet long and 40 feet wide, we can use the same formula:

Area = length x width

So, for the second shape, the area is:

Area = 60 ft x 40 ft

Area = 2,400 square feet

To find the total area of both shapes added together, we simply add the two areas:

Total Area = 16,000 square feet + 2,400 square feet

Total Area = 18,400 square feet

Therefore, the total area of both shapes added together is 18,400 square feet.

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So the polynomial 2x² + x³ - 7x + 1 in standard form is x³ + 2x² - 7x + 1.

To rewrite the polynomial 2x² + x³ - 7x + 1 in standard form, we need to write the terms in descending order of degree.

So we start with the highest degree term:

x³

Then we add the next highest degree term: 2x²

Followed by the next highest degree term: -7x

Finally, we add the constant term: +1

Putting all the terms together, we get:

x³ + 2x² - 7x + 1

So the polynomial 2x² + x³ - 7x + 1 in standard form is x³ + 2x² - 7x + 1.

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A rectangular prism is a three-dimensional shape with six faces, all of which are rectangles. The volume of a rectangular prism can be found by multiplying the length, width, and height of the prism.

In this case, Sam's box has a height of 1/6 inches, a width of 1/3 inches, and a length of 1/2 inches. To find the volume, we need to multiply these three dimensions:

(1/6) x (1/3) x (1/2) = 1/36 cubic inches.

Therefore, the volume of Sam's box is 1/36 cubic inches, which is an improper fraction because the numerator is greater than the denominator.

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the answer to log 45 will be Q + 2P.

What is Logarithmic functions?

A logarithmic function is a type of function that can be expressed in the form: f(x) = a log(bx + c) + d where a, b, c, and d are constants and x is the independent variable. The base of the logarithm is usually assumed to be 10, but can be any other positive number.

Logarithmic functions are used in a variety of applications, including finance, physics, and engineering. In finance, logarithmic functions are used to calculate compound interest. In physics, logarithmic functions are used to describe exponential decay and growth. In engineering, logarithmic functions are used to model the behavior of electrical circuits and other systems.

log a (45) = log a (9) + log a (5)

Next, we can use the fact that log a (x^n) = n log a (x) to simplify the first term:

log a (9) = log a (3²) = 2 log a (3) = 2p

Finally, we can substitute the given values for p and q and simplify the expression:

log a (45) = 2p + q = 2 log a (3) + log a (5) = log a (3²) + log a (5) = log a (3² * 5)

Therefore, we have:

log a (45) = log a (3² * 5)

Now, using the property that log a (x * y) = log a (x) + log a (y), we can simplify this expression even further:

log a (45) = log a (3²) + log a (5) = 2 log a (3) + log a (5) = Q + 2P

Therefore, log a (45) is equivalent to Q + 2P.

Therefore, the answer is Q + 2P.

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(a) Given: f(x) = x^2 + 4x + 3.

The partial fraction decomposition of f(x) is:

f(x) = (x+1)(x+3)

Now, we need to find the integral of this function from 1 to infinity:

∫[1,∞] (x+1)(x+3) dx

Since the integral converges, we can conclude that the series also converges.

(b) This series is not geometric, so we don't know what it converges to. However, we can decompose the given series as the difference of two sums:

Σ(n^2 + 4n + 3) = Σ(n^2) - Σ(4n)

(c) We can use index shifts to make these sums look similar enough to rewrite the expression without Σ:

Σ(n^2) - Σ(4n) = Σ(n^2 - 4n)

(d) To find the limit as B approaches 0, we can evaluate the limit of the expression n^2 + 4n + 3:

lim(B→0) (n^2 + 4n + 3) = n^2 + 4n + 3

So, the limit of the series is n^2 + 4n + 3.

A fraction that is greater than 1/2 and less than 4/5, we need to consider fractions between these two values. A fraction that is greater than 1/2 and less than 4/5 is 6/10.

Compare the two given fractions by finding a common denominator.
The lowest common denominator for 1/2 and 4/5 is 10.
Convert both fractions to equivalent fractions with a denominator of 10.
1/2 = 5/10
4/5 = 8/10
Identify a fraction between 5/10 and 8/10.
One possible fraction between these two is 6/10.
A fraction that is greater than 1/2 and less than 4/5 is 6/10.

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The system of inequalities in the graph is the one in option A.

y ≥ (-2/5)x - 2/5

y ≥ (3/2)*x - 1

Here we can see the graph of a system of inequalities, on the graph we can see two lines.

The first one is a line with a positive slope, it has an y-intercept of -1, the shaded region is above that line,  and it is a solid line, so one of the inequalities is:

y ≥ a*x - 1

Where a is positive.

The second line has a negative slope, and we can see that the shaded region is also above the line, so this second inequality is like:

y ≥ line with negative slope.

It is easy to identify the correct option because there is only one with these properties, which is the first option:

y ≥ (-2/5)x - 2/5

y ≥ (3/2)*x - 1

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If 400 people attended a concert 10 percent of the people came from Scotland 25 percent of the people came form Wales, there were 60 more people from Wales than from Scotland.

To find out how many more people came from Wales than Scotland at a concert with 400 attendees, we'll first calculate the number of people from each region.

1. Determine the number of people from Scotland:
Since 10% of the people came from Scotland, we'll multiply the total attendees (400) by 10% (0.10).
400 * 0.10 = 40 people from Scotland.

2. Determine the number of people from Wales:
Since 25% of the people came from Wales, we'll multiply the total attendees (400) by 25% (0.25).
400 * 0.25 = 100 people from Wales.

3. Calculate the difference between the number of attendees from Wales and Scotland:
Subtract the number of people from Scotland (40) from the number of people from Wales (100).
100 - 40 = 60 more people from Wales than Scotland.

In conclusion, at the concert with 400 attendees, there were 60 more people from Wales than from Scotland.

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In the next 20 students surveyed, you would expect 5 to own a dog and a cat

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

Dog = 42

Cat = 34

Dog and cat = 15

Neither = 9

This means that

P(Dog and cat) = 15/100

When evaluated, we have

P(Dog and cat) = 5/20

So, when the next 20 students surveyed, we have

Dog and cat = 5/20 * 20

Evaluate

Dog and cat = 5

Hence, the number of dogs and cats is 5

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Answer:

a. x = 68

b. x = 55

c. x = 18

Step-by-step explanation:

Formula

Inscribed angle = Central angle/2

a.

x = 136/2

x = 68

b.

x = ( 360 - 150 - 100 )/2

= 110/2

x = 55

c.

x = 18

Based on the linear regression formula y = 1.3485x + 840.51, you can calculate the monthly rental price (y) for a one-bedroom apartment by plugging in the square footage (x) of the apartment.

The linear regression formula for the 15 similar one-bedroom apartments is y = 1.3485x + 840.51, where x represents the square footage and y represents the monthly rental price. This means that for every square foot increase in the apartment size, the monthly rental price is predicted to increase by $1.35.

The y-intercept of the formula is $840.51, which represents the predicted monthly rental price for an apartment with 0 square footage (this is not possible in reality, but is used in the formula for mathematical purposes). To get the rental price, round your answer to the nearest whole number. For example, if an apartment has 500 square feet, you'd calculate: y = 1.3485(500) + 840.51 ≈ 1344.76, which rounds to $1,345 as the monthly rental price.

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The next time they will coincide is on November 21. when Lorena uses a social network every 8 days, Luis logs in every 6 days, and his sister and Alexa log in every 10 days.

To find the time when all three coincided, we need to find the least common multiple (LCM) of 6, 8, and 10. The LCM of 6, 8, and 10 is given as,

6 8 10 | 2

3  4  5 | 3

1   4  5 | 4

1   1  5 | 5

1   1   1

LCM = 2 × 3 × 4 × 5 = 120

if they coincided on July 24, To find the time when all three coincided we need to add 120 days to July 24 to find the next time they will coincide.  if we add 120 days to July 24 we will get the result as November 21.

Therefore, The next time they will coincide is on November 21.

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The question is,

Lorena is a student who uses a social network every 8 days. His friend Luis logs in every 6 days and his sister Alexa logs in every 10 days. If they coincided with their visit to this social network on July 24 when will they coincide next time?

Your answer is D. 1/18  is the probability that she will choose a chocolate bar and then a piece of gum

First, let's determine the total number of snacks in the bag:
4 bags of chips + 5 fruit snacks + 6 chocolate bars + 3 pieces of bubble gum = 18 snacks

Next, let's find the probability of choosing a chocolate bar:
There are 6 chocolate bars and 18 snacks total, so the probability is 6/18, which simplifies to 1/3.

Since she puts the chocolate bar back, the total number of snacks remains the same. Now, let's find the probability of choosing a piece of gum:
There are 3 pieces of gum and 18 snacks total, so the probability is 3/18, which simplifies to 1/6.

Finally, to find the probability of both events happening, multiply the probabilities together:
(1/3) * (1/6) = 1/18

So, the probability that Debra will choose a chocolate bar and then a piece of gum is 1/18. Your answer is D. 1/18.

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Cubic inches of popcorn does the jumbo size hold compared to the regular size round to the nearest tenth

To determine how many more cubic inches of popcorn the jumbo size holds compared to the regular size, you would need to:

1. Find the volume (in cubic inches) of both the jumbo and regular size popcorn containers.
2. Subtract the volume of the regular size container from the volume of the jumbo size container.

Unfortunately, without specific dimensions for the jumbo and regular size popcorn containers, I cannot provide a numerical answer. Please provide the dimensions, and I would be happy to help you with the calculations.

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The ranking from worst to best is Portfolio 3, Portfolio 2, and Portfolio 1. So, correct option is B.

To calculate the weighted mean of the RORs for each portfolio, we need to first multiply each ROR by the corresponding portfolio value and then sum the products for each portfolio. We then divide the total by the sum of the portfolio values.

The weighted mean for Portfolio 1 = [(10.4% x $700) + (-29.7% x $12,000) + (37.2% x $600) + (7.5% x $4,400) + (6.3% x $250)] / ($700 + $12,000 + $600 + $4,400 + $250) = -16.8%

Similarly, the weighted mean for Portfolio 2 = 3.8% and for Portfolio 3 = 11.2%.

Based on the results, the list that shows a comparison of the overall performance of the portfolios from worst to best is option b) Portfolio 3, Portfolio 2, Portfolio 1. Portfolio 3 has the highest weighted mean return of 11.2%, followed by Portfolio 2 with a return of 3.8%, and Portfolio 1 has the lowest weighted mean return of -16.8%.

Therefore, correct option is B.

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Complete question is:

ROR     Portfolio 1     Portfolio 2     Portfolio 3

10.4%        $700              $6,000         $3,500

-29.7%    $12,000           $9,000         $5,500

37.2%        $600              $4,500         $5,750

7.5%          $4,400            $2,000        $1,500

6.3%          $250                $1,100          $4,500

Calculate the weighted mean of the RORs for each portfolio. Based on the results, which list shows a comparison of the overall performance of the portfolios, from worst to best?

a) Portfolio 3, Portfolio 1, Portfolio 2

b) Portfolio 3, Portfolio 2, Portfolio 1

c) Portfolio 1, Portfolio 2, Portfolio 3

d) Portfolio 1, Portfolio 3, Portfolio 2

The volume of the rain barrel is approximately 10,256.58 cubic inches.

To get the volume of the rain barrel, which is shaped like a right circular cylinder, you need to use the formula for the volume of a cylinder: V = πr²h. Here, V represents the volume, r is the radius, and h is the height of the cylinder.
The given diameter of the rain barrel is 22 inches. To find the radius (r), you need to divide the diameter by 2:
r = 22 / 2 = 11 inches.
The height (h) of the rain barrel is given as 27 inches.
Now, you can plug these values into the formula and use 3.14 for pi (π):
V = πr²h
V = 3.14 * (11²) * 27
V = 3.14 * (121) * 27
V = 3.14 * 3267
V ≈ 10,256.58 cubic inches
So, the volume of the rain barrel is approximately 10,256.58 cubic inches.

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Add the given numbers: -8 + 4 + (-2) = -6. So, the sum of -8, 4, and -2 is -6.

To represent -6 on a number line, we need to find its position relative to zero. Since -6 is negative, it will be located to the left of zero. We count 6 units to the left of zero on the number line to represent -6. Therefore, the number line that shows the sum of -8, 4, and -2 is:

o----+----+----+----+----+----+----+----+----+----o

-15 -10 -5 0 5 10 15 20 25 30

-6

So, the complete answer is:

The sum of -8, 4, and -2 is -6.

To represent -6 on a number line, locate 6 units to the left of zero.

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Answer:

(Credit to guy/girl above) 63 miles 10 1/2 x 6 is 63.

Step-by-step explanation:

pls mark brainliest

When the manufacturer sells 100,000 cell phones, her revenue will be approximately $1,657,993.39.

To find the revenue for selling 100,000 cell phones, we will first evaluate both binomials for the given number of cell phones (c = 100,000) and then multiply them together.

Step 1: Evaluate the first binomial (number of cell phones sold) for c = 100,000:
0.015c + 2.81 = 0.015(100,000) + 2.81 = 1,500 + 2.81 = 1,502.81

Step 2: Evaluate the second binomial (wholesale price) for c = 100,000:
0.011c + 3.52 = 0.011(100,000) + 3.52 = 1,100 + 3.52 = 1,103.52

Step 3: Calculate the revenue by multiplying the results of the two binomials:
Revenue = (1,502.81)(1,103.52) = 1,657,993.3912

So, when the manufacturer sells 100,000 cell phones, her revenue will be approximately $1,657,993.39. This calculation is based on the binomial expressions provided for the number of cell phones sold (0.015c+2.81) and the wholesale price (0.011c+3.52).

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The vocabulary for the segment a is the apothem

The area of the polygon is about 41.6 yd²

The area of a regular figure is the extent of the planer space the figure occupies.

The length of each side of the regular polygon, s = 4 yd

The length of the segment a = 2·√3

The vocabulary term for the segment a drawn from from the center of the polygon and perpendicular to one of its sides is the apothem

Therefore, the vocabulary term for segment a is the apothem

The polygon is a hexagon.

The area of a hexagon is; A = ((3·√3)/2) × s²

Therefore, the area of the polygon is; A = ((3·√3)/2) × (4)² = 24·√3 ≈ 41.6

The area of the polygon is about 41.6 yd²

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The indicated real nth root(s) of a, where n=3 and a=27 is 3.


You need to find the indicated real nth root(s) of a, where n=3 and a=27. In other words, you need to find the real number(s) that, when raised to the power of 3, equal 27.

Here's a step-by-step explanation:

1. Identify the given values: n=3 and a=27.
2. Write the equation: x^n = a, where x is the real nth root you're trying to find.
3. Substitute the given values: x^3 = 27.
4. Solve the equation for x: x = 3, since 3^3 = 27.

Your answer is x = 3, which is the real 3rd root of 27.

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You can look the image.

It is very clear.

Answer:

[tex]\sin (Z)=\sf\dfrac{9}{15}[/tex]        [tex]\cos (Z)=\sf\dfrac{12}{15}[/tex]        [tex]\tan(Z)=\sf \dfrac{9}{12}[/tex]

Step-by-step explanation:

To create trigonometric ratios for angle Z in the given right triangle XYZ, we can use the trigonometric ratios.

[tex]\boxed{\begin{minipage}{9.4 cm}\underline{Trigonometric ratios} \\\\$\sf \sin(\theta)=\dfrac{O}{H}\quad\cos(\theta)=\dfrac{A}{H}\quad\tan(\theta)=\dfrac{O}{A}$\\\\where:\\ \phantom{ww}$\bullet$ $\theta$ is the angle. \\ \phantom{ww}$\bullet$ $\sf O$ is the side opposite the angle. \\\phantom{ww}$\bullet$ $\sf A$ is the side adjacent the angle. \\\phantom{ww}$\bullet$ $\sf H$ is the hypotenuse (the side opposite the right angle). \\\end{minipage}}[/tex]

From inspection of the right triangle XYZ:

Substitute these values into the three ratios to create the trigonometric ratios for angle Z:

[tex]\sin (Z)=\sf \dfrac{O}{H}=\dfrac{9}{15}[/tex]

[tex]\cos (Z)=\sf \dfrac{A}{H}=\dfrac{12}{15}[/tex]

[tex]\tan(Z)=\sf \dfrac{O}{A}=\dfrac{9}{12}[/tex]

The surface area of pyramid B to the nearest hundredth is 58.67 cm²

Similar figures are two figures having the same shape. The ratio of the corresponding sides of similar shapes are equal.

The scale ratio of the height of the pyramid A to B is

9/6 = 3/2

Area factor = (3/2)² = 9/4

9/4 = 132/x

9x = 132×4

9x = 528

divide both sides by 9

x = 528/9

x = 58.67cm²

Therefore the surface area of pyramid B is 58.67cm².

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Knowing that tan(x) = 3/5 and using a trigonometric identity, we will get that:

tan(2x) = 1.875

There is a trigonometric identity we can use for this, we know that:

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

So we only need to knos tan(x), which we already know that is equal to 3/5, then we can replace it in the formula above to get:

[tex]tan(2x) = \frac{2*3/5}{1 - (3/5)^2}\\\\tan(2x) = \frac{6/5}{1 - 9/25} \\tan(2x) = 1.875[/tex]

That is the value of the tangent of 2x.

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The size of three angles of the triangle are 55 degrees, 55 degrees, and 15 degrees. The three consecutive odd numbers are 9, 11, and 13 and three consecutive even numbers are 32, 34, and 36.

1.To find the size of each angle in the triangle, we know that the sum of all angles in a triangle is 180 degrees. So we can set up an equation:

(2x + 5) + (2x + 5) + (x - 10) + 65 = 180

Simplifying and solving for x, we get:

5x + 55 = 180

5x = 125

x = 25

Now we can substitute x back into the expressions for each angle and simplify:

2x + 5 = 55 degrees

2x + 5 = 55 degrees

x - 10 = 15 degrees

Therefore, the three angles of the triangle are 55 degrees, 55 degrees, and 15 degrees.

2. Let's call the first odd number x. Then the next two consecutive odd numbers would be x + 2 and x + 4. We know that the sum of these three numbers is 33, so we can set up an equation:

x + (x + 2) + (x + 4) = 33

Simplifying and solving for x, we get:

3x + 6 = 33

3x = 27

x = 9

Therefore, the three consecutive odd numbers are 9, 11, and 13.

3. Let's call the first even number x. Then the next two consecutive even numbers would be x + 2 and x + 4. We know that the sum of these three numbers is 102, so we can set up an equation:

x + (x + 2) + (x + 4) = 102

Simplifying and solving for x, we get:

3x + 6 = 102

3x = 96

x = 32

Therefore, the three consecutive even numbers are 32, 34, and 36.

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Gabby does not have any fabric left, as she used more fabric than she had to start with.

How much fabric does Gabby have left after using 3215 yards, and what is the estimate?

Based on the information given, Gabby started with 578 yards of fabric and used 3215 yards of fabric. To estimate the amount of fabric Gabby has left, we need to subtract the amount of fabric used from the starting amount of fabric:

578 yards - 3215 yards = -2637 yards

Since the result is a negative number, it doesn't make sense in this context. It's possible that there was a mistake in the numbers given, or Gabby used more fabric than she had to start with.

Without further information or clarification, we cannot estimate the amount of fabric Gabby has left.

To estimate the amount of fabric Gabby has left, we can subtract the amount of fabric she used from the amount of fabric she had initially.

So, to find the estimate for the amount of fabric Gabby has left, we can perform the following calculation:

Estimate for the amount of fabric Gabby has left = 578 yards (initial amount) - 3215 yards (amount used)

Estimate for the amount of fabric Gabby has left = -2637 yards

However, the result is negative, which means that Gabby doesn't have any fabric left, and she needs to purchase an additional 2637 yards to make up for the shortfall.

Therefore, the estimate for the amount of fabric Gabby has left is 0 yards (she needs to purchase more fabric to continue her work).

The area of the ring-shaped region with radii of 9m and 13m is approximately 276.32 square meters.

What is Area?

The area is the region defined by an object's shape. The area of a shape is the space covered by a figure or any two-dimensional geometric shape in a plane.

What is Perimeter?

The perimeter of a shape is defined as the total distance surrounding the shape. It is the length of any two-dimensional geometric shape's outline or boundary.

According to the given information:

The given shape is a two concentric circles with radii of 9m and 13m, we can calculate the area of this region using the formula for the area of a circle:

Area of shaded region = Area of outer circle - Area of inner circle

The area of a circle is given by the formula A = πr^2, where r is the radius of the circle.

Area of inner circle = π(9)^2 = 81π

Area of outer circle = π(13)^2 = 169π

Area of shaded region = 169π - 81π = 88π

Using the value of π = 3.14, we get:

Area of shaded region = 88π = 88(3.14) = 276.32 square meters (rounded to two decimal places)

Therefore, the area of the ring shaped region with radii of 9m and 13m is approximately 276.32 square meters.

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We can see that the absolute minimum occurs at (x, y) = (2, -48).

To find the relative and absolute extrema of the function g(x) = 3x³ - 36x on the domain [-4, 4], we first need to find the critical points. We do this by finding the first derivative, setting it to zero, and solving for x.

g'(x) = d(3x³ - 36x)/dx = 9x² - 36

Setting g'(x) to 0:

0 = 9x² - 36
x² = 4
x = ±2

These are our critical points. To determine if these are minima, maxima, or neither, we use the second derivative test.

g''(x) = d(9x² - 36)/dx = 18x

At x = -2:
g''(-2) = -36 < 0, so it's a relative maximum.

At x = 2:
g''(2) = 36 > 0, so it's a relative minimum.

Now, we need to compare the function values at the critical points and endpoints of the domain to determine the absolute extrema.

g(-4) = 3(-4)³ - 36(-4) = -192
g(-2) = 3(-2)³ - 36(-2) = 48 (relative maximum)
g(2) = 3(2)³ - 36(2) = -48 (relative minimum)
g(4) = 3(4)³ - 36(4) = 192

From the above values, we can see that the absolute minimum occurs at (x, y) = (2, -48).

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A. The distribution function F(x,y) is ¹¹/₁₈ + ¹/₁₂ x² - ¹/₁₈ y² + ¹/₁₂ xy

B. The marginal PDF of x is ¹/₂x + ¹/₆ + ¹/₁₂x² for 0≤x≤1 and for y is /₂y + ¹/₆ + ¹/₁₂y² for 0≤y≤2

C. P(0≤x≤1/2, 0≤y≤1/2) is ¹/₃₂ + ¹/₉₆ x² for 0≤x≤1/2

D. The joint PDF of u=2x-y and v=-x+y is f(u,v) = (1/27)(2u^2+2v^2-2uv)

a) To find the distribution function F(x,y), integrate the joint PDF over the appropriate limits.

F(x,y) = ∫∫f(u,v)dudv

The limits of integration are not specified, so, determine them from the limits of the variables x and y.

So,

F(x,y) = ∫∫f(u,v)dudv

= ∫∫f(x+y,x-y)dudv (substituting u = x+y and v = x-y)

= ∫∫(¹/₄(u²+v²)+¹/₆(u²-v²))dudv (substituting x and y back in terms of u and v)

The limits of integration for u and v can be found by solving for u and v in terms of x and y as follows:

u = x+y

v = x-y

x = (u+v)/2

y = (u-v)/2

0 ≤ x ≤ 1; 0 ≤ y ≤ 2

implies

0 ≤ (u+v)/2 ≤ 1; 0 ≤ (u-v)/2 ≤ 2

Solving the above inequalities gives the following limits:

0 ≤ u ≤ 2; -u ≤ v ≤ u;

Thus,

F(x,y) = ∫∫(¹/₄(u²+v²)+¹/₆(u²-v²))dudv

= ∫²₀ ∫ᵘ_(-u) (1/4(u²+v²)+¹/₆(u²-v²))dvdu

= ¹¹/₁₈ + ¹/₁₂ x² - ¹/₁₈ y² + ¹/₁₂ xy

b) To find the marginal PDF of x, integrate the joint PDF over all possible values of y:

f(x) = ∫f(x,y)dy

So,

f(x) = ∫²₀ (¹/₄x + ¹/₄y²/x + ¹/₆y) dy

= ¹/₂x + ¹/₆ + ¹/₁₂x² for 0≤x≤1

In the same way, find the marginal PDF of y, by integrating the joint PDF over all possible values of x:

f(y) = ∫f(x,y)dx

So,

f(y) = ∫¹₀ (¹/₄x²/y + ¹/₄y + ¹/₆xy) dx

= ¹/₂y + ¹/₆ + ¹/₁₂y² for 0≤y≤2

c) To find P(0≤x≤1/2, 0≤y≤1/2), integrate the joint PDF over the appropriate limits:

P(0≤x≤1/2, 0≤y≤1/2) = ∫∫f(x,y)dxdy

So,

P(0≤x≤1/2, 0≤y≤1/2) = ∫¹₀ ∫^(1/2)_0 (¹/₄x² + ¹/₄y²/x + ¹/₆xy) dydx

= ¹/₃₂ + ¹/₉₆ x² for 0≤x≤1/2

d) To find the joint PDF of u=2x-y and v=-x+y, express x and y in terms of u and v and then apply transformation formula.

From the given equations, solve for x and y in terms of u and v as follows:

x = (u+v)/3

y = (v-u)/3

Now, find the Jacobian of the transformation:

J = ∂(x,y)/∂(u,v) =

| ∂x/∂u ∂x/∂v |

| ∂y/∂u ∂y/∂v |

=

| 1/3 1/3 |

| -1/3 1/3 |

So, |J| = 2/9

Using the transformation formula for joint PDFs:

f(u,v) = f(x(u,v), y(u,v)) |J|

Substituting x and y in terms of u and v:

f(u,v) = f((u+v)/3, (v-u)/3) (2/9)

Substituting the given joint PDF for f(x,y), we get:

f(u,v) = (¼((u+v)/3)² + ¼((v-u)/3)² + ⅙((u+v)/3)((v-u)/3))(2/9)

Simplify:

f(u,v) = (1/27)(2u²+2v²-2uv)

So, the joint PDF of u=2x-y and v=-x+y is:

f(u,v) = (1/27)(2u²+2v²-2uv)

learn more about distribution function: https://brainly.com/question/25301796

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