A bakery records the number of cakes, x it makes and the corresponding total price, p, of the cakes, in dollars. Number of Cakes (x) Price (p) 1 12 2 24 3 36 4 48 Write an equation that represents the relationship between x and p?

The given differential equation dy/dx = sin(x)/y is a first-order separable differential equation.

A separable differential equation is one that can be expressed in the form g(y)dy = f(x)dx, where g(y) and f(x) are functions of y and x, respectively. In this case, we have dy/dx = sin(x)/y, which can be rewritten as ydy = sin(x)dx.

To solve this separable differential equation, we can integrate both sides:

∫ydy = ∫sin(x)dx

Integrating the left side with respect to y gives (1/2)y^2, and integrating the right side with respect to x gives -cos(x) + C, where C is the constant of integration.

Therefore, we have (1/2)y^2 = -cos(x) + C.

The separable differential equation dy/dx = sin(x)/y can be solved by integrating both sides. The solution is given by (1/2)y^2 = -cos(x) + C, where C is the constant of integration.

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Thus, the equation that connects C and X is C = 100X + 5.33X² and the depth of the well if the cost is 12000 naira is 38.85 meters.

1. Deriving an equation that connects C and X together The cost C of sinking a well X meters deep varies partly as X and partly X². That is,C = kX + pX²,Where k and p are constants to be determined. To determine the value of k and p, we can use the information given that the cost is 5000 naira if the depth is 30m and cost is 8000 naira if the depth is 50m.From the above information, we can get two equations:

5000 = 30k + 30²p8000 = 50k + 50²p

We can use the first equation to get the value of k and substitute it in the second equation.

5000 = 30k + 900p ⇒ k = 166.67 - 10p

Substituting k in the second equation gives:

8000 = 50(166.67 - 10p) + 2500p

Solving the above equation gives:

p = 5.33 And, k = 100.00

Substituting k and p in the cost equation gives:

C = 100X + 5.33X²2. Finding the depth of the well if the cost is 12000 naira

Given that C = 12000, we need to find the value of X.C = 100X + 5.33X² ⇒ 5.33X² + 100X - 12000 = 0

Solving the above quadratic equation using the quadratic formula gives:

X = (-b ± √(b²-4ac))/2a = (-100 ± √(100² - 4×5.33×(-12000)))/2×5.33= (-100 ± 540.71)/10.66= 38.85 or -23.45

'Since the depth can't be negative, the depth of the well is X = 38.85 meters when the cost is 12000 naira.

Thus, the equation that connects C and X is C = 100X + 5.33X² and the depth of the well if the cost is 12000 naira is 38.85 meters.

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To find the percentage of current net income that Juniper must set aside for new bills, we can use the following formula:

percent increase = (new price - old price) / old price * 100%

In this case, the old price is 585 ,and the new price is 600. To calculate the percentage increase, we can use the formula above:

percent increase = (600−585) / 585∗100

percent increase = 15/585 * 100%

percent increase = 0.0263 or approximately 2.63%

To find the percentage of current net income that Juniper must set aside for new bills, we can use the following formula:

percent increase = (new price - old price) / old price * 100% * net income

where net income is Juniper's current net income after setting aside the percentage of her income for new bills.

Substituting the given values into the formula, we get:

percent increase = (600−585) / 585∗100

= 15/585 * 100% * net income

= 0.0263 * net income

To find the percentage of current net income that Juniper must set aside for new bills, we can rearrange the formula to solve for net income:

net income = (old price + percent increase) / 2

net income = (585+15) / 2

net income =600

Therefore, Juniper must set aside approximately 2.63% of her current net income of 600 for new bills.

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a) The determinant of A is non-zero, the vectors v1, v2, and v3 are linearly independent and do not lie in a plane in R3.

b) The determinant of B is non-zero, the vectors v1, v2, and v3 are linearly independent and do not lie in a plane in R3.

To determine whether three vectors lie in a plane in R3, we need to check if they are linearly dependent or independent. If they are linearly dependent, then they lie in a plane; if they are linearly independent, then they do not lie in a plane.

(a) To check if v1, v2, and v3 lie in a plane, we need to see if they are linearly dependent or independent. One way to do this is to find the determinant of the matrix A whose columns are the three vectors:

| 2  6  2 |

|−2  1  0 |

| 0  4 −4 |

We can expand this determinant along the first row to get:

det(A) = 2 × | 1  0 |

       - (-2) × | 6  4 |

       + 0 × | 1 −4 |

       = 2(1 × 4 - 0 × (-4)) - (-2)(6 × 4 - 1 × 1) + 0

       = 8 + 47 + 0

       = 55

(b) To check if v1, v2, and v3 lie in a plane, we need to see if they are linearly dependent or independent. One way to do this is to find the determinant of the matrix B whose columns are the three vectors:

|−6  3  4 |

| 7  2 −1 |

| 2  4  2 |

We can expand this determinant along the third column to get:

det(B) = 4 × |−6  3 |

       - (-1) × | 7  2 |

       + 2 × | 2  4 |

       = 4(-6 × 2 - 3 × 7) - (-1)(7 × 4 - 2 × 2) + 2(2 × 2 - 4 × 3)

       = -96 + 30 + (-8)

       = -74

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To write and solve a story problem with 6 divided by 6, we need to come up with a situation in which 6 is divided equally among 6 parts. For example:

There are 6 pieces of candy to be divided equally among 6 children. Solution: To solve this problem, we can simply divide the total number of candies (6) by the number of children (6):6 ÷ 6 = 1Therefore, each child will receive 1 piece of candy. Another way to solve this problem is to use multiplication. Since division is the inverse of multiplication, we can think of this problem as:6 ÷ 6 = x can be rewritten as 6 = x × 6, where x is the number of candies each child receives. Then we can solve for x by dividing both sides by 6:x = 6 ÷ 6x = 1Therefore, each child will receive 1 piece of candy.

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The volume of the cone frustum is 4.19 cubic units.

To find the volume of the cone frustum, we can use the formula:

[tex]V = (1/3)\pi h(R^2 + Rr + r^2)[/tex]

where h is the height of the frustum, R and r are the radii of the top and bottom bases, respectively.

In this case, the frustum is given by the inequality[tex]z = 2x^2 + y^2 < 2[/tex] and is bounded by the planes z = 0 and z = 3. This means that the height of the frustum is h = 3 - 0 = 3.

To find the radii R and r, we need to find the intersection of the cone [tex]z = 2x^2 + y^2[/tex] and the plane z = 2. Substituting z = 2 into the cone equation, we get:

[tex]2 = 2x^2 + y^2[/tex]

This is the equation of an ellipse in the xy-plane with major axis along the x-axis and minor axis along the y-axis.

To find the radii, we can use the standard form of the ellipse:

[tex](x/a)^2 + (y/b)^2 = 1[/tex]

where a and b are the semi-major and semi-minor axes, respectively. Comparing this with the equation of the ellipse above, we get:

[tex]a^2 = 1/2[/tex] and [tex]b^2 = 2[/tex]

Therefore, the radii are R = √(1/2) and r = √2.

Substituting these values into the formula for the volume, we get:

V = (1/3)π(3)(1/2 + √2/2 + 2)

Simplifying this expression, we get:

V = (π/3)(√2 + 5)

Therefore, the volume of the cone frustum is approximately 4.19 cubic units.

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Answer:

C & D

Step-by-step explanation:

Sonali purchased some pants and skirts the numbers of skirts is 7 less than eight times the number of pants purchase also number of skirt is four less than five times the number of pants purchased is 1 pant and 1 skirt.

Let's denote the number of pants Sonali purchased as P and the number of skirts as S. We're given two pieces of information:

1. The number of skirts (S) is 7 less than eight times the number of pants (8P). This can be represented as S = 8P - 7.

2. The number of skirts (S) is also 4 less than five times the number of pants (5P). This can be represented as S = 5P - 4.

Now we have a system of two linear equations with two variables, P and S:

S = 8P - 7
S = 5P - 4

To solve the system, we can set the two expressions for S equal to each other:

8P - 7 = 5P - 4

Solving for P, we get:

3P = 3
P = 1

Now that we know P = 1, we can substitute it back into either equation to find S. Let's use the first equation:

S = 8(1) - 7
S = 8 - 7
S = 1

So, Sonali purchased 1 pant and 1 skirt.

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The global maximum value of f(x,y) on the triangular region is 9, which occurs at (2,0), and the global minimum value is -14, which occurs at (0,3).

To find the global maximum and minimum values of the function f(x,y) = 1 + 4x - 5y on the closed triangular region with vertices (0,0), (2,0), and (0,3), we need to evaluate the function at each vertex and on each line segment connecting the vertices, and then compare the values.

First, let's evaluate f(x,y) at each vertex:

f(0,0) = 1 + 4(0) - 5(0) = 1

f(2,0) = 1 + 4(2) - 5(0) = 9

f(0,3) = 1 + 4(0) - 5(3) = -14

Next, let's evaluate f(x,y) on each line segment connecting the vertices:

On the line segment connecting (0,0) and (2,0):

y = 0, so f(x,0) = 1 + 4x

f(1,0) = 1 + 4(1) = 5

On the line segment connecting (0,0) and (0,3):

x = 0, so f(0,y) = 1 - 5y

f(0,1) = 1 - 5(1) = -4

f(0,2) = 1 - 5(2) = -9

f(0,3) = -14

On the line segment connecting (2,0) and (0,3):

y = -5/3x + 5, so f(x,-5/3x + 5) = 1 + 4x - 5(-5/3x + 5)

Simplifying this expression, we get f(x,-5/3x + 5) = 21/3x - 24/3

f(1,2/3) = 1 + 4(1) - 5(2/3) = 19/3

f(0,3) = -14

Therefore, the global maximum value of f(x,y) on the triangular region is 9, which occurs at (2,0), and the global minimum value is -14, which occurs at (0,3).

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a) To find t0.005 when v = 6, we need to look up the value in a t-distribution table with a two-tailed area of 0.005 and 6 degrees of freedom. From the table, we find that t0.005 = -3.707.

b) To find t0.025 when v = 11, we need to look up the value in a t-distribution table with a two-tailed area of 0.025 and 11 degrees of freedom. From the table, we find that t0.025 = -2.201.

c) To find t0.99 when v = 18, we need to look up the value with a one-tailed area of 0.99 and 18 degrees of freedom. From the table, we find that t0.99 = 2.878. Note that we only look up one-tailed area since we are interested in the value in the upper tail of the distribution.

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For the sample population

(a) The point estimate of p is 0.71.

(b) Using Equation 73-2, the approximate 90% confidence interval for p is obtained by calculating 0.71 ± 1.645 * sqrt((0.71 * (1 - 0.71))/200).

(c) Using Equation 73-4, the approximate 90% interval for p is found by calculating 0.71 ± 1.645 * sqrt((0.71 * (1 - 0.71))/(200 - 1)).

(d) Using Equation 73-5, the approximate 90% confidence interval for p is obtained by calculating 0.71 ± 1.645 * sqrt((0.71 * (1 - 0.71))/(200 + 1.645^2/4)).

(a) To obtain a point estimate of p, we divide the number of letters delivered the next day (y = 142) by the sample size (n = 200):

Point estimate of p = y/n = 142/200 = 0.71

(b) Using Equation 73-2, we can find an approximate 90% confidence interval for p. The formula is given by:

Point estimate ± Z * sqrt((p * (1 - p))/n)

Since the confidence level is 90%, the Z-value for a 90% confidence level is approximately 1.645. Substituting the values into the equation:

Confidence interval = 0.71 ± 1.645 * sqrt((0.71 * (1 - 0.71))/200)

Simplifying the expression:

Confidence interval = 0.71 ± 1.645 * sqrt(0.21/200)

(c) Using Equation 73-4, we can find an approximate 90% interval for p. The formula is given by:

Point estimate ± Z * sqrt((p * (1 - p))/(n - 1))

Applying the formula with the given values:

Confidence interval = 0.71 ± 1.645 * sqrt((0.71 * (1 - 0.71))/(200 - 1))

Simplifying the expression:

Confidence interval = 0.71 ± 1.645 * sqrt(0.21/199)

(d) Using Equation 73-5, we can find an approximate 90% confidence interval for p. The formula is given by:

Point estimate ± Z * sqrt((p * (1 - p))/(n + Z^2/4))

Substituting the values into the equation:

Confidence interval = 0.71 ± 1.645 * sqrt((0.71 * (1 - 0.71))/(200 + 1.645^2/4))

Simplifying the expression:

Confidence interval = 0.71 ± 1.645 * sqrt(0.21/200.5084)

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The measure of angle C in triangle CDE is 9 degrees

To find the measure of angle C in triangle CDE, we need to solve the given equation.

The measure of angle C is (x - 4) degrees.

In the triangle, the sum of the measures of all three angles must be equal to 180 degrees (since it is a triangle). So we can set up the equation:

(x - 4) + (11x - 11) + (x + 13) = 180

Simplifying the equation:

2x - 4 + 11x - 11 + x + 13 = 180

14x - 2 = 180

14x = 182

x = 13

Substituting x = 13 into the equation for angle C:

(x - 4) = (13 - 4) = 9

Therefore, the measure of angle C is 9 degrees.

In summary, the measure of angle C in triangle CDE is 9 degrees. To find this value, we set up an equation using the sum of the measures of all three angles in a triangle, and then solved for x by simplifying and rearranging the equation. Substituting the value of x into the equation for angle C gives us the final answer of 9 degrees.

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the simplified expression for y ″ is (390y^2) / (4x^3).

To find y ″ by implicit differentiation, we need to differentiate both sides of the given equation with respect to x twice, using the chain rule and product rule as needed.

First, we differentiate both sides of x^2 5y^2 = 5 with respect to x using the product rule:

d/dx (x^2 5y^2) = d/dx (5)

Using the product rule, we get:

(2x)(5y^2) + (x^2)(d/dx (5y^2)) = 0

Simplifying and using the chain rule, we get:

10xy^2 + 2x^2y(dy/dx) = 0

Next, we differentiate both sides of this equation with respect to x again, using the product rule and chain rule as needed:

d/dx (10xy^2 + 2x^2y(dy/dx)) = d/dx (0)

Using the product rule and chain rule, we get:

10y^2 + 20xy(dy/dx) + 2x^2(dy/dx)^2 + 2x^2y(d^2y/dx^2) = 0

Simplifying and solving for d^2y/dx^2, we get:

d^2y/dx^2 = (-10y^2 - 4x^2(dy/dx)^2) / (4xy)

To simplify this expression, we need to find an expression for dy/dx. We can use the original equation to do this:

x^2 5y^2 = 5

Differentiating both sides with respect to x using the chain rule, we get:

2x(5y^2) + (x^2)(d/dx (5y^2)) = 0

Simplifying and using the chain rule, we get:

10xy + 2x^2y(dy/dx) = 0

Solving for dy/dx, we get:

dy/dx = -10y/x

Substituting this expression into the expression we found for d^2y/dx^2, we get:

d^2y/dx^2 = (-10y^2 - 4x^2((-10y/x)^2)) / (4xy)

Simplifying, we get:

d^2y/dx^2 = (-10y^2 + 400y^2) / (4x^3)

d^2y/dx^2 = (390y^2) / (4x^3)

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To find the corresponding values of x, we need to solve the equation 2x = 2 pi/3 and 2x = 8 pi/3 for x over the interval [0, 2 pi).

So, the corresponding values of x are x = π/3, π, 4π/3.

To find the corresponding values of x for the given trigonometric equations, we need to divide each equation by 2:
1. For 2x = 2π/3, divide by 2:
            x = (2π/3) / 2

               = π/3

2. For 2x = 8π/3, divide by 2:
            x = (8π/3) / 2

               = 4π/3

Taking the given interval,
3. For 2x = 2π, divide by 2:
            x = 2π / 2

               = π

Hence, the solution for the values of x are π/3, π, 4π/3.

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The length of the radius rounded to 2 decimal places is 4.88 cm.

To find the length of the radius of a circle given its area, you can use the formula:

Area = π * radius²

Given that the area is 74.8 cm², we can set up the equation:

74.8 = π * radius²

To solve for the radius, we need to rearrange the equation and isolate the radius:

radius² = 74.8 / π

radius = √(74.8 / π)

Now, let's calculate the value using a calculator:

radius ≈ √(74.8 / 3.14159)

radius ≈ √23.7839769

radius ≈ 4.876

Rounded to 2 decimal places, the length of the radius is approximately 4.88 cm.

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The circle with a center at (0, 0) and passing through (0, -3) has an area and circumference that can be calculated. The area can be found using the formula A = πr^2, and the circumference can be found using the formula C = 2πr, where r is the radius of the circle.

Given that the center of the circle is at (0, 0) and it passes through (0, -3), we can determine that the radius of the circle is 3 units. The distance between the center (0, 0) and the point on the circle (0, -3) gives us the radius.

To find the area of the circle, we use the formula A = πr^2. Substituting the radius, we have A = π(3^2) = 9π square units.

To find the circumference of the circle, we use the formula C = 2πr. Substituting the radius, we have C = 2π(3) = 6π units.

Therefore, the area of the circle is 9π square units, and the circumference of the circle is 6π units.

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The PMF of Y=max{0,X} is P(Y=k) = (b-k+1)/(b-a+1) for k = 0,1,2,...,b and P(Y=k) = 0 for all other values of k.

The PMF of W=min{0,X} is P(W=k) = (k-a+1)/(b-a+1) for k = a,a+1,a+2,...,0 and P(W=k) = 0 for all other values of k. This is because for Y, the probability of X taking a certain value decreases as that value gets larger, but for W, the probability of X taking a certain value increases as that value gets more negative.

Therefore, the PMF for Y will have a peak at k=0 and decrease as k increases, while the PMF for W will have a peak at k=a and decrease as k becomes more negative.

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A derivative is a mathematical concept that represents the rate at which a function is changing at a given point. It is a measure of how much a function changes in response to a small change in its input.

We can start by finding the first derivative of the function:

f(x) = x^2 - 4x

f'(x) = 2x - 4

Then, we can find the second derivative:

f''(x) = d/dx (2x - 4) = 2

So, f''(2) = 2.

the value of f''(2) is 2.

what is function?

In mathematics, a function is a relation between a set of inputs and a set of possible outputs with the property that each input is related to exactly one output. A function is typically represented by an equation or rule that assigns a unique output value for each input value.

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The inverse variation equation is y = k/x where k is the constant of proportionality; when x = 2, y = 16.

y = k/x

Where,

k = constant of proportionality

When y = 4; x = 8

y = k/x

4 = k/8

k = 4 × 8

k = 32

When x = 2

y = k/x

y = 32/2

y = 16

Hence, the value of y when x = 2 is 16

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A = ($120× 12× 45)[tex](1+0.067/12)^{(12*45)}[/tex]

This is the final amount Samantha would have in the savings account after 45 years.

To calculate the amount of money Samantha would have in the savings account after 45 years, we can use the formula for compound interest:

A = P[tex](1+r/n)^{nt}[/tex]

Where:

A = the final amount of money

P = the principal amount (initial investment)

r = annual interest rate (in decimal form)

n = number of times the interest is compounded per year

t = number of years

In this case:

P = $120 per month

r = 6.7% = 0.067 (decimal form)

n = 12 (compounded monthly)

t = 45 years

First, we need to calculate the total amount invested over 45 years. Since Samantha invests $120 per month, the total amount invested would be:

Total Amount Invested = $120/month× 12 months/year ×45 years

Next, we can calculate the final amount using the compound interest formula:

A = P[tex](1+r/n)^{nt}[/tex]

A = ($120 × 12 × 45)[tex](1+0.067/12)^{(12*45)}[/tex]

Calculating this expression will give us the final amount Samantha would have in the savings account after 45 years.

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To solve this problem, we need to subtract the depth at which the dolphin is located from the sea level.What is a depth?Depth refers to the distance from the surface to the bottom of a body of water or any other object.

To put it another way, depth is a measurement of distance from the surface of something downward or inward.For example, when an object, say a Dolphin, is at a depth of 45 3/4 feet relative to sea level, how many feet has it descended from sea level?We must perform the following calculation to get our answer:45 3/4 feetSo, the dolphin has descended 45 3/4 feet from sea level.

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If f is continuous and ∫(14f(x)dx) from 0 to 6 = 6, then ∫(7f(2x)dx) from 0 to 3 = 3.

To explain this, let's follow these steps:

1. We are given that ∫(14f(x)dx) from 0 to 6 = 6.

2. Divide both sides of the equation by 2 to get ∫(7f(x)dx) from 0 to 6 = 3.

3. Now, apply the substitution method: let u = 2x, so du/dx = 2 and dx = du/2.

4. Change the limits of integration: when x = 0, u = 2(0) = 0; when x = 3, u = 2(3) = 6.

5. Substitute u into the integral and adjust the limits: ∫(7f(u)du/2) from 0 to 6.

6. The constant 7/2 can be factored out of the integral: (7/2)∫(f(u)du) from 0 to 6.

7. Since we know that ∫(7f(x)dx) from 0 to 6 = 3, we can conclude that (7/2)∫(f(u)du) from 0 to 6 = 3.
8. So, ∫(7f(2x)dx) from 0 to 3 = 3.

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The solution for 20 degrees in the first quadrant is:

20 degrees = 20π/180 = 0.349 radians.

Starting with the given equation:

7cos(20) + 3 = sin(20) + 4

Rearranging:

7cos(20) - sin(20) = 1

Using the trig identity cos(a-b) = cos(a)cos(b) + sin(a)sin(b):

cos(20-70) = cos(-50) = cos(50)

Using the fact that cosine is an even function:

cos(50) = cos(-50)

So we can write:

cos(50) = 1/7

Therefore, the solution for 20 degrees in the first quadrant is:

20 degrees = 20π/180 = 0.349 radians.

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To find out how long it will take for the egg to hit the ground, we need to determine the value of t when the height (h) of the egg is zero. In other words, we need to solve the equation:

-16t^2 + 30 = 0

To solve this quadratic equation, we can use the quadratic formula:

t = (-b ± √(b^2 - 4ac)) / (2a)

In this case, a = -16, b = 0, and c = 30. Substituting these values into the quadratic formula, we get:

t = (± √(0^2 - 4*(-16)30)) / (2(-16))

Simplifying further:

t = (± √(0 - (-1920))) / (-32)

t = (± √1920) / (-32)

t = (± √(64 * 30)) / (-32)

t = (± 8√30) / (-32)

Since time cannot be negative in this context, we can disregard the negative solution. Therefore, the time it will take for the egg to hit the ground is:

t = 8√30 / (-32)

Simplifying this further, we get:

t ≈ -0.791 seconds

The negative value doesn't make sense in this context since time cannot be negative. Therefore, we discard it. So, the egg will hit the ground approximately 0.791 seconds after being dropped.

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The arithmetic average return is found by adding up the returns and dividing by the number of years:

Arithmetic average = (22% + 9% - 7% + 13%) / 4 = 9.25%

To find the geometric average return, we need to use the formula:

Geometric average = (1 + R1) x (1 + R2) x ... x (1 + Rn) ^ (1/n) - 1

where R1, R2, ..., Rn are the annual returns.

So for this stock, the geometric average return is:

Geometric average = [(1 + 0.22) x (1 + 0.09) x (1 - 0.07) x (1 + 0.13)] ^ (1/4) - 1

                  = 0.0868 or 8.68%

Therefore, the arithmetic average return is 9.25% and the geometric average return is 8.68%.

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Warren sold 330 cars last month. He can now calculate his commission based on the commission rate he is paid for the month.

Warren is paid commission based on the number of cars he sells. To calculate his commission, he needs to know how many cars he sold last month. The following table shows the data he recorded in the log book for the month: Car Sales Log Book Car Sales Car Sales Car Sales Day 1Day 2Day 3Day 4Day 5Day 6Day 7Day 8Day 9Day 102010 2020 3030 4040 3030 5050 6060 4040 2020We can see that on Day 1, Warren sold 20 cars, and on Day 2, he sold 20 cars. On Day 3, he sold 30 cars, and on Day 4, he sold 40 cars.

On Day 5, he sold 30 cars, and on Day 6, he sold 50 cars. On Day 7, he sold 60 cars, and on Day 8, he sold 40 cars. Finally, on Day 9, he sold 20 cars, and on Day 10, he sold 20 cars.

The total number of cars Warren sold for the month can be calculated by adding up the number of cars sold each day: Total number of cars sold = 20 + 20 + 30 + 40 + 30 + 50 + 60 + 40 + 20 + 20 = 330 cars Therefore, Warren sold 330 cars last month. With this information, he can now calculate his commission based on the commission rate he is paid for the month.

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Answer:

The change is exponential growth and the percent increase is 57.3%

Step-by-step explanation:

An exponential growth function is represented by the equation

f(x)=a(1+r)^t

As such r is equal to 0.573, or 57.3%

The answer is 4/507.

Using AM-GM inequality, we have:

x^2y^2/13 = (x^2/13) (y^2/13) (169/169) ≤ ((x^2/13) + (y^2/13) + (169/169))/3 = (x^2 + y^2 + 169)/507

Since x + y = 2, we have x^2 + y^2 ≥ 2xy = 4 - 2y, so:

x^2 + y^2 + 169 ≥ 173 - 2y

Thus, x^2y^2/13 ≤ (173 - 2y)/507 for any nonnegative x and y with x + y =

2. This expression is a decreasing function of y, so its maximum value occurs at y = 0 and its minimum value occurs at y = 2. Thus:

Max: (173 - 2(0))/507 = 173/507

Min: (173 - 2(2))/507 = 169/507

The difference between these is:

173/507 - 169/507 = 4/507

So the answer is 4/507.

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The surface is given by the equations x = 5t, y = 4sin(u), and z = 4cos(u)

To find a parametric representation for the surface, we can start by introducing the variables u and v.

Let u and v be parameters representing the angles around the y and z-axes respectively.

Then, we can express y and z in terms of u and v as follows:

y = 4sin(u) z = 4cos(u)

Since x is bounded between 0 and 5, we can express x in terms of another parameter t as x = 5t, where 0 < t < 1.

Combining the equations for x, y, and z, we obtain the parametric representation: x = 5t y = 4sin(u) z = 4cos(u)

Thus, the surface is given by the equations x = 5t, y = 4sin(u), and z = 4cos(u), where 0 < t < 1 and 0 ≤ u ≤ 2π.

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The value of length ED in the cuboid is determined as 8.7 cm.

The value of length ED is calculated as follows;

The line connecting point E to point D is a diagonal line, and the magnitude is calculated by applying Pythagoras theorem as follows;

ED² = AE² + AD²

From the diagram, AE = CG = 8.5 cm,

also, length AD = BC = 2 cm

The value of length ED is calculated as;

ED² = 8.5² + 2²

ED = √ ( 8.5² + 2²)

ED = 8.7 cm

Thus, the length of ED is determined by applying Pythagoras theorem as shown above.

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