I don’t understand how to do this

Answer: x +

Step-by-step explanation:

the correct expression is:

5 x (8 + 4)

when expanded, this would give you:

5x8 + 5x4

so the blanks should be filled with x and +

a) The solution to the initial-value problem is: y^3 = 3xr^3 + 2.

b) The solution to the initial-value problem is: x^2 y + (1/2) y^3 = (1/2) r^2 + 3/2.

c) The solution to the initial-value problem is: xr - ye^y = z ln|x| + 1.

(a) We can start by separating the variables and integrating both sides with respect to x and y:
xy^2 dy = y^3 - r^3 dx
Integrating both sides:
(1/3) y^3 = xr^3/3 + C
Using the initial condition y(2) = 2:
(1/3) (2)^3 = 2r^3/3 + C
C = 2/3
Thus, the solution to the initial-value problem is:
y^3 = 3xr^3 + 2

(b) Similar to part (a), we can separate the variables and integrate both sides with respect to x and y:
x^2 + 2y^2 dy = ry dx
Integrating both sides:
x^2 y + (1/2) y^3 = (1/2) r^2 + C
Using the initial condition y(-1) = 1:
(-1)^2 (1) + (1/2) (1)^3 = (1/2) r^2 + C
C = 3/2
Thus, the solution to the initial-value problem is:
x^2 y + (1/2) y^3 = (1/2) r^2 + 3/2

(c) We can start by multiplying both sides by dx and integrating:
(x-yey) dr = zey dy/x
Integrating both sides:
xr - ye^y = z ln|x| + C
Using the initial condition y(1) = 0:
1r - 0e^0 = z ln|1| + C
C = 1
Thus, the solution to the initial-value problem is:
xr - ye^y = z ln|x| + 1

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Answer:

Step-by-step explanation:

Solve for x.

4x - 9 = 2x + 5

4x - 2x = 5 + 9

2x = 14

x = 14 : 2

x = 7

-----------------

check   (replace "x" with "7")

4 * 7 - 9 = 2 * 7 + 5                  (remember PEMDAS)

28 - 9 = 14 + 5

19 = 19

the answer is good

Answer:

hence the required value of x is 7.

Answer:

Step-by-step explanation:

Answer:

(4, -1)

Hope you can understand what I did

a. The correlation matrix p =  [tex]\left[\begin{array}{ccc}1&1/3&2/5\\1/3&1&-3/5\\2/5&-3/5&1\end{array}\right][/tex]. b. The correlation between X1, and i/2X2 + 1/2X3 is 0.3.

a. The correlation matrix p can be calculated by dividing the covariance matrix by the product of the standard deviations of the variables:

p = [tex]\left[\begin{array}{ccc}1&1/3&2/5\\1/3&1&-3/5\\2/5&-3/5&1\end{array}\right][/tex]

b. To compute the correlation between X1 and i/2X2 + 1/2X3, we first need to calculate the standard deviations of the variables:

σ1 = sqrt(4) = 2

σ2 = sqrt(9) = 3

σ3 = sqrt(25) = 5

Then, we can calculate the covariance between X1 and i/2X2 + 1/2X3:

cov(X1, i/2X2 + 1/2X3) = cov(X1, i/2X2) + cov(X1, 1/2X3)

= i/2 * cov(X1, X2) + 1/2 * cov(X1, X3)

= i/2 * 1 + 1/2 * 2

= 1.5

Finally, we can compute the correlation using the formula:

corr(X1, i/2X2 + 1/2X3) = cov(X1, i/2X2 + 1/2X3) / (σ1 * σ2/2 + σ3/2)

= 1.5 / (2 * 3/2 + 5/2)

= 0.3

Therefore, the correlation between X1 and i/2X2 + 1/2X3 is 0.3.

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The study aimed to provide insights into the potential effects of rhGH and rhIGF-I, both separately and in combination, on the immune function of elderly individuals, as indicated by the immune response to tetanus toxoid immunization.

The study aimed to evaluate the impact of a 7-week administration of recombinant human growth hormone (rhGH) and recombinant human insulin-like growth factor (rhIGF-I), both separately and in combination, on immune function in elderly female rhesus monkeys. The researchers used the response to immunization with tetanus toxoid as an assay to measure the in vivo function of the immune system.

The study design likely involved the following steps:

Selection of elderly female rhesus monkeys as the study subjects: The researchers chose female monkeys of advanced age to represent the elderly population.

Administration of recombinant human growth hormone (rhGH): The researchers administered rhGH to a group of monkeys for a period of 7 weeks. This hormone is known to stimulate growth and metabolism.

Administration of recombinant human insulin-like growth factor (rhIGF-I): Another group of monkeys received rhIGF-I, a hormone that mediates the effects of GH, for the same duration.

Combination treatment: A third group of monkeys received both rhGH and rhIGF-I simultaneously during the 7-week period.

Immunization with tetanus toxoid: After the 7-week treatment period, all monkeys were immunized with tetanus toxoid, which is a vaccine used to induce an immune response against tetanus.

Measurement of immune response: The researchers assessed the immune function by measuring the response of the monkeys' immune systems to the tetanus toxoid immunization. They likely examined parameters such as antibody production or T-cell response.

Data analysis: The researchers analyzed the immune response data to determine the effects of rhGH, rhIGF-I, and their combination on the immune function of the elderly female rhesus monkeys.

The study aimed to provide insights into the potential effects of rhGH and rhIGF-I, both separately and in combination, on the immune function of elderly individuals, as indicated by the immune response to tetanus toxoid immunization.

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The population of Pennsylvania is: 1304503 people

Population density is calculated by taking the total area of a region in question and dividing it by the total number of people that live in that area. The result will give the average number of inhabitants per square kilometre, mile, acre, meter, etc.

The parameters given are:

Population of colorado = 5,770,000 people

Area of colorado = 280 * 380

= 106,400 mi²

Population density here = 5,770,000/106,400

54.23 people per mi²

Area of Pennsylvania = 283 * 170

= 48110 mi²

Thus:

Population of Pennsylvania/48110 mi² = 5 * 54.23 people per mi²

Population of Pennsylvania = 48110 * mi² * 5 * 54.23 people per mi²

= 1304503 people

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Answer:

The perimeter of the quarter circle is 24.997 cm

Step-by-step explanation:

Given, the radius of the circle = 7 cm

The perimeter of the circle = 2πr

and perimeter of the quarter circle = 2r + C

where r is the radius and C is the circumference of the sector of a circle

Circumference of the sector =  ∅/360°(2πr)

                                           C  = 90°/360°(2×3.142×7)

                                            C = 10.997 cm

perimeter of the quarter circle = 2r + C  

                                                   = 2×7 + 10.997

                                                   = 24.997 cm

The perimeter of the quarter circle will be 24.997 cm

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Answer:9.5

Step-by-step explanation:

Answer: 8

Step-by-step explanation:

The median of this data set is 8. If you cross one number from both sides at the same time, you will eventually come to the middle of the data set, which is 8.

Therefore, if Isabel plans to drive inequality more than 25 miles, Company B will be more expensive than Company A. If she plans to drive 25 miles or less, Company A will be more expensive.

Let's start by setting up an inequality to represent the mileages for which Company A charges less than Company B.

For Company A, the cost is a flat fee of $90, regardless of the number of miles driven.

For Company B, the cost depends on the number of miles driven. The initial fee is $75, and then there is an additional charge of $0.60 for every mile driven. So, the total cost for Company B can be represented by the equation:

Cost(B) = 0.60m + 75

here m is the number of miles driven.

We want to find the mileages for which Company A charges less than Company B. In other words, we want to find the values of m for which:

Cost(A) < Cost(B)

Substituting in the expressions for the costs, we get:

90 < 0.60m + 75

Simplifying and solving for m, we get:

15 < 0.60m

25 < m

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The absolute value equation that has a solution set of 3 and 7 is |x-5|= 2.

We have,

The solution sets of the absolute value equation are given as x = {3, 7}.

So, Mean of solution

x₁= (7+3)/2

= 10/2

= 5

and, x₂ = (7-3)/2

= 4/2

= 2

Now, the absolute value equation

|x - x₁| - x₂ = 0

|x -5|-2 = 0

|x-5|= 2

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A proportional relationship is a type of linear relationship where the ratio of two variables is constant.

The number of hours studied (x) and the corresponding grade on a test (y) for a group of students. We observe that the grades are directly proportional to the number of hours studied. To write an equation to describe this proportional relationship, we can use the form y = kx, where k is the constant of proportionality.

To find the value of k, we can use any data point in the dataset. Let's say that when a student studies for 5 hours, they get a grade of 80. We can substitute these values into the equation:

[tex]80 = k[/tex] × [tex]5[/tex]

To solve for k, we can divide both sides by 5:

[tex]k = 80 / 5[/tex]

[tex]= 16[/tex]

Therefore, the equation to describe this proportional relationship is:

[tex]y = 16x[/tex]

This means that for every additional hour studied, the grade increases by 16 points.

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Complete Question:

How to write an equation to describe a proportional relationship?

We are 98% confident that the mean of the true population y lies between 10.8 and 13.4.

To construct the confidence interval, we first need to calculate the critical value of t using the given values of C and n. Since C = 0.98, we can find the level of significance as α = 1 - C = 0.02.

Using a t-table or calculator, the critical value of t for a two-tailed test with 14 degrees of freedom and

[tex]\frac{\alpha}{2} = 0.01[/tex] is approximately 2.977.

Next, we can calculate the sample standard deviation as s = σ/√n = [tex]\frac{0.95}{\sqrt{15}}= 0.245[/tex].

Then, we can use the formula for a confidence interval for the population mean using the t-distribution:
(y ± t)×[tex]\frac{s}{\sqrt{n}}[/tex]

Substituting the given values, we get:

(12.1 ± 2.977)×[tex]\frac{0.245}{\sqrt{15}}[/tex]

Simplifying and rounding to one decimal place, we get the confidence interval: (10.8, 13.4)

Therefore, we are 98% confident that the true population mean y lies between 10.8 and 13.4.

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The solution of the differential equation 4y'' − y = [tex] {e}^{x/2} [/tex] + 8 by variation of parameter method is y(x) = (15C - 16)[tex] {e}^{x/2} [/tex] + 15C' [tex] {ex}^{-x/2} [/tex]

To solve the differential equation by variation of parameters, we assume that the solution is of the form,

y(x) = u₁(x)y₁(x) + u₂(x)y₂(x), linearly independent solutions of the homogeneous equation are y₂(x) and y₂(x), and functions to be determined u₁(x) and u₂(x). The homogeneous equation associated with the given differential equation is,

4y'' - y = 0

The characteristic equation is,

4r² - 1 = 0 which has solutions r = ±1/2. Therefore, the general solution of the homogeneous equation is,

y(x) = C[tex] {e}^{x/2} [/tex] + C'[tex] {e}^{-x/2} [/tex]

C and C' are arbitrary constants.

Now, we need to find particular solutions of the non-homogeneous equation. We can guess that a particular solution has the form,

[tex] y_{p(x)} = A(x) {e}^{(x/2)} [/tex]

where A(x) is a function to be determined. We can find A(x) by substituting y_p(x) into the differential equation and solving for A(x). We have,

[tex] 4y_{p(x)} - y_{p(x)} = {e}^{(x/2)} +8 [/tex]

Differentiating twice and substituting these into the differential equation gives:

[tex]4( A"(x) + A'(x)) {e}^{2/y} 2 + \frac{A(x)}{4} - A(x) {e}^{(x/2)} = {e}^{(x/2)} + 8[/tex]

Simplifying and solving for A(x), we obtain,

A(x) = -16/15

Therefore, a particular solution of the differential equation is:

[tex]y_{p(x)} = \frac{ - 16}{15} {e}^{(x \div 2)} [/tex]

The general solution of the non-homogeneous equation is then,

y(x) = C[tex] {e}^{x/2} [/tex] + C'[tex] {e}^{-x/2} [/tex] [tex]\frac{ - 16}{15} {e}^{(x/2)} [/tex]

Simplifying and collecting terms, we get,

y(x) = (15C - 16)[tex] {e}^{x/2} [/tex] + 15C' [tex] {ex}^{-x/2} [/tex] ,where C and C' are arbitrary constants.

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Complete question - Solve the differential equation by variation of parameters. 4y'' − y = e^x/2 + 8.

The table which contains only values that satisfy the equation of line defined as y = 0. 5x + 14, ( linear equation) is present in option(c). So, option(c) is right one.

We have a equation of line, y = 0.5x + 14, --(1) which is a equation of line . We have to recognise the table which satisfy the above line equation. The values are called roots of the equation. A value that is a solution of an equation is said to satisfy the equation, and the solutions of an equation create its solution set. The above table consists values of x and y, so we check which set of values form solution set of equation (1). Let x = 0 => y = 14 so, ( 0, 14) is solution of equation (1). Similarly, when x = 5

=> y = 5× 0.5 + 14 = 16.5

Similarly, we can check other point values. The table present in option (c) is correct answer.

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Complete question:

The above figure complete the question.

The coupon rate for the semi-annual bond is approximately 11.96%.

Given the information provided, we have the following details:

- Par value: $1,000
- Maturity: 10 years
- Current price: $887
- YTM (Yield to Maturity): 10.9%

To solve for the coupon rate, we can use the bond pricing formula:

Bond Price = (C * (1 - (1 + r/2)^(-2n))) / (r/2) + (Par Value / (1 + r/2)^(2n))

Where:
- Bond Price = $887
- C = Coupon payment per period (which we need to find)
- r = YTM / 100 = 0.109
- n = Maturity in years = 10

Plugging in the given values:

$887 = (C * (1 - (1 + 0.109/2)^(-2*10))) / (0.109/2) + ($1,000 / (1 + 0.109/2)^(2*10))

Now, we can solve for the coupon payment, C:

C = (($887 * 0.109/2) - ($1,000 / (1 + 0.109/2)^(2*10))) / (1 - (1 + 0.109/2)^(-2*10))

C ≈ $59.80

Since this is a semi-annual bond, the annual coupon payment would be:

Annual Coupon Payment = C * 2 = $59.80 * 2 = $119.60

Finally, to find the coupon rate, we can divide the annual coupon payment by the par value:

Coupon Rate = (Annual Coupon Payment / Par Value) * 100 = ($119.60 / $1,000) * 100 = 11.96%

So, the coupon rate for the semi-annual bond is approximately 11.96%.

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the dimensions of the rectangle are L = 15 inches and W = 15 inches, and the minimum perimeter is:    P = 2L + 2W = 60 inches.

Let the length and width of the rectangle be L and W, respectively, so that the area of the rectangle is A = LW = 225. We want to find the dimensions of the rectangle with minimum perimeter P = 2L + 2W, and then find the minimum perimeter.

Using the given area, we can solve for one of the variables in terms of the other:

L = 225/W

Substituting this expression for L into the expression for the perimeter, we get:

P = 2(225/W) + 2W

Taking the derivative of P with respect to W and setting it equal to zero to find the minimum, we get:

[tex]dP/dW = -450/W^2 + 2 = 0[/tex]

Solving for W, we get:

W^2 = 225

Since W must be positive (it is a length), we take the positive square root:

W = 15

Substituting this value of W back into the expression for L, we get:

L = 225/W = 15

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The three options with dimensions resulting in a surface area greater than 140 square feet are:

To determine whether the dimensions of a rectangular prism result in a surface area greater than 140 square feet, we can use the formula for the surface area of a rectangular prism:

Surface Area = 2lw + 2lh + 2wh

    where l, w, and h are the length, width, and height of the rectangular prism, respectively.

Option 1: 6 feet by 2 feet by 3 feet

Surface Area = 2(6)(2) + 2(6)(3) + 2(2)(3) = 24 + 36 + 12 = 72 square feet

This option does not have a surface area greater than 140 square feet.

Option 2: 6 feet by 5 feet by 4 feet

Surface Area = 2(6)(5) + 2(6)(4) + 2(5)(4) = 60 + 48 + 40 = 148 square feet

This option has a surface area greater than 140 square feet.

Option 3: 7 feet by 6 feet by 4 feet

Surface Area = 2(7)(6) + 2(7)(4) + 2(6)(4) = 84 + 56 + 48 = 188 square feet

This option has a surface area greater than 140 square feet.

Option 4: 8 feet by 3 feet by 7 feet

Surface Area = 2(8)(3) + 2(8)(7) + 2(3)(7) = 48 + 112 + 42 = 202 square feet

This option has a surface area greater than 140 square feet.

Option 5: 8 feet by 4 feet by 3 feet

Surface Area = 2(8)(4) + 2(8)(3) + 2(4)(3) = 64 + 48 + 24 = 136 square feet

This option does not have a surface area greater than 140 square feet.

Therefore, the three options with dimensions resulting in a surface area greater than 140 square feet are:

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As per the given equation, the stone was dropped from a height of approximately 1.08 feet or about 13 inches.

The velocity of a free-falling object is an important concept in physics, and it is defined by the equation:

v = 2gh

In this equation, v represents the velocity of the object in feet per second, g represents the acceleration due to gravity in feet per second per second, and h represents the distance that the object has fallen in feet.

Suppose a stone is dropped from a certain height and strikes the water with a velocity of 138 feet per second. Our task is to estimate the height from which the stone was dropped.

To solve this problem, we need to rearrange the equation to solve for h. We start by dividing both sides of the equation by 2g:

h = v/2g

Substituting the given values, we get:

h = 138/2(32) = 1.078125

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To determine the supremum and infimum of the given sets.

(a) The set {1/n: n ∈ N} consists of the reciprocals of positive integers. The smallest element in the set is 1, as it corresponds to n=1. The set has no largest element since it has an infinite number of elements getting smaller as n increases. Therefore, the infimum (greatest lower bound) of the set is 1, and there is no maximum. The supremum (least upper bound) of the set is not in the set itself, but it exists and equals 1.

(b) The set {z ∈ Q: 22 < 3} is an empty set since there is no rational number z that satisfies the condition 22 < 3. In this case, there is no supremum or infimum since the set has no elements. Consequently, there is no maximum or minimum value.

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The quadratic equation with zeros at -6 and 2 is y² + 4y - 12 = 0. This is in standard form, which is ax² + bx + c = 0, with a = 1, b = 4, and c = -12.

To find the quadratic equation with zeros at -6 and 2, we can start by using the fact that if a quadratic equation has roots x₁ and x₂, then it can be written in the form

(y - x₁)(y - x₂) = 0

where y is the variable in the quadratic equation.

Substituting the given values of the zeros, we get

(y - (-6))(y - 2) = 0

Simplifying this expression, we get

(y + 6)(y - 2) = 0

Expanding this expression, we get

y² - 2y + 6y - 12 = 0

Simplifying this expression further, we get

y² + 4y - 12 = 0

So the quadratic equation with zeros at -6 and 2 is

y² + 4y - 12 = 0

This is the standard form of a quadratic equation, which is

ax² + bx + c = 0

where a, b, and c are constants. In this case, a = 1, b = 4, and c = -12.

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The relation that is a function is given as follows:

Bottom left graph.

A graph represents a function if it has no vertically aligned points, that is, each value of x is mapped to only one value of y. Vertically aligned points mean that a value of x is mapped to multiple values of y, that is, a single input is mapped to multiple outputs which disqualify the relation as a function.

Hence the bottom left graph is the only one with a relation representing a function, as a vertical line would not cross the graph of the function more than once no matter where it was plotted.

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I)

[tex]~~~~~~~~~~~~\textit{middle point of 2 points } \\\\ A(\stackrel{x_1}{1}~,~\stackrel{y_1}{5})\qquad B(\stackrel{x_2}{3}~,~\stackrel{y_2}{9}) \qquad \left(\cfrac{ x_2 + x_1}{2}~~~ ,~~~ \cfrac{ y_2 + y_1}{2} \right) \\\\\\ \left(\cfrac{ 3 +1}{2}~~~ ,~~~ \cfrac{ 9 +5}{2} \right) \implies \left(\cfrac{ 4 }{2}~~~ ,~~~ \cfrac{ 14 }{2} \right)\implies (2~~,~~7)[/tex]

II)

keeping in mind that perpendicular lines have negative reciprocal slopes, let's check for the slope of the line AB

[tex](\stackrel{x_1}{1}~,~\stackrel{y_1}{5})\qquad (\stackrel{x_2}{3}~,~\stackrel{y_2}{9}) ~\hfill \stackrel{slope}{m}\implies \cfrac{\stackrel{\textit{\large rise}} {\stackrel{y_2}{9}-\stackrel{y1}{5}}}{\underset{\textit{\large run}} {\underset{x_2}{3}-\underset{x_1}{1}}} \implies \cfrac{ 4 }{ 2 } \implies 2 \\\\[-0.35em] ~\dotfill[/tex]

[tex]\stackrel{~\hspace{5em}\textit{perpendicular lines have \underline{negative reciprocal} slopes}~\hspace{5em}} {\stackrel{slope}{ 2 \implies \cfrac{2}{1}} ~\hfill \stackrel{reciprocal}{\cfrac{1}{2}} ~\hfill \stackrel{negative~reciprocal}{-\cfrac{1}{2} }}[/tex]

so we're really looking for the equation of a line whose slope is -1/2 and it passes through (2 , 7)

[tex](\stackrel{x_1}{2}~,~\stackrel{y_1}{7})\hspace{10em} \stackrel{slope}{m} ~=~ - \cfrac{1}{2} \\\\\\ \begin{array}{|c|ll} \cline{1-1} \textit{point-slope form}\\ \cline{1-1} \\ y-y_1=m(x-x_1) \\\\ \cline{1-1} \end{array}\implies y-\stackrel{y_1}{7}=\stackrel{m}{- \cfrac{1}{2}}(x-\stackrel{x_1}{2}) \\\\\\ y-7=- \cfrac{1}{2}x+1\implies {\Large \begin{array}{llll} y=- \cfrac{1}{2}x+8 \end{array}}[/tex]

The solution to the given differential equation is:

[tex]$\frac{1}{2}x^4y + \frac{2}{3}x^3y^2 = C$[/tex]

To solve the given differential equation:

[tex]$(x^2 + y^2 + xy)dx + (xy)dy = 0$[/tex]

We will first check whether this is an exact differential equation or not.

[tex]$\frac{\partial M}{\partial y} = \frac{\partial }{\partial y}(x^2 + y^2 + xy) = 2y + x$[/tex]

[tex]$\frac{\partial N}{\partial x} = \frac{\partial }{\partial x}(xy) = y$[/tex]

Since [tex]$\frac{\partial M}{\partial y} \neq \frac{\partial N}{\partial x}$[/tex], this is not an exact differential equation.

Next, we will check if this differential equation is solvable using an integrating factor.

[tex]$\frac{1}{xy}(x^2 + y^2 + xy)dx + dy = 0$[/tex]

Let us assume the integrating factor as [tex]$u = u(x)$[/tex]

Multiplying both sides of the differential equation with the integrating factor, we get:

[tex]$\frac{1}{y}(x^2 + y^2 + xy)u(x)dx + u(x)dy = 0$[/tex]

Now, we can see that this is an exact differential equation.

[tex]$\frac{\partial }{\partial y}\left(\frac{1}{y}(x^2 + y^2 + xy)u(x)\right) = \frac{xu(x)}{y}$[/tex]

[tex]$\frac{\partial }{\partial x}\left(u(x)\right) = \frac{xu(x)}{y}$[/tex]

Solving this differential equation, we get:

[tex]$\ln |u(x)| = \frac{1}{2}\ln(x^2y^2) = \ln(xy)$[/tex]

[tex]$u(x) = xy$[/tex]

Multiplying the integrating factor to the original differential equation, we get:

[tex]$(x^3y + x^2y^2 + x^2y^2)dx + (x^2y^2)dy = 0$[/tex]

[tex]$(x^3y + 2x^2y^2)dx + (x^2y^2)dy = 0$[/tex]

This is now an exact differential equation and can be solved by finding the potential function:

[tex]$\frac{\partial }{\partial x}\left(\frac{1}{2}x^4y + \frac{2}{3}x^3y^2\right) = x^3y + 2x^2y^2$[/tex]

[tex]$\frac{\partial }{\partial y}\left(\frac{1}{2}x^4y + \frac{2}{3}x^3y^2\right) = x^2y^2$[/tex]

Therefore, the potential function is [tex]$\frac{1}{2}x^4y + \frac{2}{3}x^3y^2 = C$[/tex], where C is the constant of integration.

Hence, the solution to the given differential equation is:

[tex]$\frac{1}{2}x^4y + \frac{2}{3}x^3y^2 = C$[/tex]

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How much does it take to cover the cylinder? is different from the rest, as it refers to the surface area of the cylinder, not its volume or capacity.

The term "cover" usually means to place something over the top or on the surface of something else, such as a lid covering a container.

In the context of a cylinder, "covering" would typically refer to finding the surface area of the cylinder, which includes both the top and bottom circles as well as the curved lateral surface.

In contrast, the other statements are related to the volume or capacity of the cylinder, which refers to how much space is contained inside the cylinder.

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The pizza parlor offers 2^48, or approximately 281 trillion, different pizzas.

To calculate the total number of different pizzas offered, we need to consider all possible combinations of pizza sizes and toppings.

For each pizza size, there are 12 options for toppings (including no toppings). Therefore, the total number of pizzas for a single size is 2^12 (2 options for each of the 12 toppings). Since there are four different sizes, the total number of different pizzas offered by the parlor is:

2^12 x 2^12 x 2^12 x 2^12 = 2^(12+12+12+12) = 2^48

Thus, the pizza parlor offers 2^48, or approximately 281 trillion, different pizzas.

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