Divide a Quadratic by a Linear

Divide the following polynomials. Express your answer in the following way:
Dividend = (Divisor)(Quotient) + Remainder

According to the popular legend, Archimedes discovered a principle of buoyancy while taking a bath.

what was his discovery ?

He realized that the volume of water displaced by his body was equal to the volume of his body. This insight led him to formulate the principle of buoyancy, which states that an object placed in a fluid experiences an upward force equal to the weight of the fluid it displaces.

Archimedes was reportedly so excited by this discovery that he ran through the streets of Syracuse shouting "Eureka!" (meaning "I have found it" in Greek). This principle has numerous practical applications, including in shipbuilding and the design of submarines and other underwater vehicles.

In geometric terms, the principle of buoyancy can be explained using the concept of displacement. When an object is submerged in a fluid, it displaces a certain volume of fluid, which is equal to the volume of the object. This volume of fluid has weight, and the fluid exerts an upward force on the object that is equal to the weight of the displaced fluid. This is what causes objects to float in water or other fluids, as long as their weight is less than the weight of the fluid they displace.

As for what Archimedes did next with his new discoveries, it's difficult to say for certain, as there are many stories and legends about his life. However, we do know that he was a prolific inventor and mathematician, and he likely continued to explore the implications of his principle of buoyancy and other discoveries. He may have also used this principle in his work as an engineer and designer, creating new machines and devices that were more efficient and effective thanks to his insights into fluid mechanics.

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We can use the quadratic formula to solve for x, and the system of equations to solve for y. We have two possible solutions for x: x = 4 + [tex]\sqrt{28}[/tex] ≈ 8.29, and y = 7 - x ≈ -0.29.

A system of linear equations can be solved using the substitution approach. In this procedure, one variable in one of the equations is solved in terms of the other, and that expression is then replaced into the other equation. This yields an equation with a single variable that can be solved. Once one variable has been identified, it may be swapped into one of the original equations to determine the second variable.

We can use the first equation to solve for one of the variables in terms of the other, and then substitute that expression into the second equation to get an equation in just one variable. Let's solve the first equation for y:

x + y = 7

y = 7 - x

Now we can substitute this expression for y into the second equation:

[tex]y = x^2 - 9x - 5[/tex]

[tex]7 - x = x^2 - 9x - 5[/tex]

Rearranging terms, we get:

[tex]x^2 - 8x - 12 = 0[/tex]

Now we can use the quadratic formula to solve for x:

x = (-b ± [tex]\sqrt{(b^2 - 4ac)}[/tex]) / 2a

where a = 1, b = -8, and c = -12. Plugging these values in, we get:

x = (-(-8) ± [tex]\sqrt{(-8)^2 - 4(1)(-12))}[/tex]) / 2(1)

Simplifying:

x = (8 ± [tex]\sqrt{(64 + 48)}[/tex]) / 2

x = (8 ± [tex]\sqrt{(112)}[/tex]) / 2

x = 4 ± [tex]\sqrt{(28)}[/tex]

So we have two possible solutions for x:

x = 4 + [tex]\sqrt{(28)}[/tex] ≈ 8.29

x = 4 - [tex]\sqrt{(28) }[/tex]≈ -0.29

Now we can use either equation to solve for y. Let's use y = 7 - x:

When x = 4 + [tex]\sqrt{(28)}[/tex], we have:

y = 7 - (4 + [tex]\sqrt{(28)}[/tex]) = 3 - [tex]\sqrt{(28)}[/tex]

When x = 4 - [tex]\sqrt{(28)}[/tex], we have:

y = 7 - (4 - [tex]\sqrt{(28)}[/tex]) = 3 + [tex]\sqrt{(28)}[/tex]

So the two solutions to the system of equations are:

x = 4 + [tex]\sqrt{(28)}[/tex], y = 3 - [tex]\sqrt{(28)}[/tex]

x = 4 - [tex]\sqrt{(28)}[/tex], y = 3 + [tex]\sqrt{(28)}[/tex]

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Answer:

23°

Step-by-step explanation:

46 = 2x (Vertically opp. angles)

x = 46÷2

x = 23

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Answer:

x = 23°

Step-by-step explanation:

46 = 2x

46/2 = 2x/2

x = 23°

1) the midpoint of the line segment is (2, 2.5). 2) the slope of the line is -3.25.  3)the length of the line segment is approximately 13.60.    4) the equation of the line is y = -3.25x + 9.

what is slope ?

In mathematics, the slope of a line is a measure of its steepness, defined as the ratio of the vertical change (rise) between two points on the line to the horizontal change (run) between the same two points. It is often denoted by the letter "m".

In the given question,

To find the midpoint of the line segment that connects the points (4,-4) and (0,9), we use the midpoint formula:

Midpoint = ((x₁ + x₂)/2, (y1 + y₂)/2)

Midpoint = ((4 + 0)/2, (-4 + 9)/2)

Midpoint = (2, 2.5)

So the midpoint of the line segment is (2, 2.5).

To find the slope of the line that passes through the two given points, we use the slope formula:

Slope = (y₂ - Ly₁)/(x₂ - x₁)

Slope = (9 - (-4))/(0 - 4)

Slope = 13/-4

Slope = -3.25

So the slope of the line is -3.25.

To find the length of the line segment that connects the two given points, we use the distance formula:

Length = √((x₂ - x₁)² + (y₂ - y1)²)

Length = √((0 - 4)² + (9 - (-4))²)

Length = √(16 + 169)

Length = √(185)

Length ≈ 13.60

So the length of the line segment is approximately 13.60.

Now, we can use the point-slope formula to find the equation of the line that passes through the point (4,-4) with slope -3.25:

y - y1 = m(x - x₁)

y - (-4) = -3.25(x - 4)

y + 4 = -3.25x + 13

y = -3.25x + 9

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Answer:

65

Step-by-step explanation:

The fraction for 6 and 10 is 6/10, which can be simplified to 3/5 by dividing both the numerator and denominator by their greatest common factor, which in this case is 2.

So 6/10 is equivalent to 3/5.

To find the middle term of the expansion of (2x/y^3)^12, we need to first determine the number of terms in the expansion.

Using the formula for the binomial theorem, we know that the number of terms in the expansion of (2x/y^3)^12 is 13.

Now, to find the middle term, we need to find the term that is in the middle of the expansion. Since there are 13 terms in the expansion, the middle term is the 7th term.

The general term of the expansion is given by:

T(r+1) = (12 C r) (2x)^r (y^-3)^(12-r)

So, the 7th term is:

T(8) = (12 C 7) (2x)^7 (y^-3)^(5)

Simplifying this expression, we get:

T(8) = (792) (128x^7 / y^15)

Therefore, the middle term of the expansion is 792(128x^7/y^15)

Answer:

You can start by changing 12 3/4 into a decimal, which would be 12.75.

The formula for perimeter of a rectangle is (l + w) x 2, so you could write it like this:

(12.75 + 6.4) x 2

Then solve the question in the parenthesis...

19.15 x 2 = 38.30

Now, the question says to write the perimeter into simplest fraction form.

38.30 = 38 3/10.

So, there you have it!

38 3/10 meters

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The associative law holds in all three composition tables.

No matter how they are arranged, integers can be multiplied and added together according to the associative property. We refer to the numbers in the parenthesis as grouping (). Let's imagine that you are adding three integers, 2, 5, and 6. It doesn't matter how we add the numbers together—using the formulas 2 + (5 + 6) or 2 + (5) + 6—the outcome will be the same. The multiplication equation is consistent: Equals to (2 x 5) x 6 is (2 x 5) x 6. The commutative characteristic is almost identical to this one when only two integers are used.

In the given composition tables the law of associativity is followed

a(bc) = (ab)c

from 13,

a(a) = bc

a = a

The associative law holds in all three composition tables

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The sum of the matrices A + B is  5      8

                                                        2.6   2

To add two matrices, they must have the same dimensions (the same number of rows and the same number of columns).

To add two matrices A and B, you can add the corresponding elements of the two matrices and place the result in the corresponding position of a new matrix C.

In mathematical notation, if A and B are two matrices of the same dimensions, then the sum C = A + B is defined by:

C[i,j] = A[i,j] + B[i,j]

Using the above as a guide, we have the following:

A + B =  5      8

             2.6   2

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Answer:

[tex]\textsf{A.} \quad \dfrac{1}{4}[/tex]

Step-by-step explanation:

Given geometric sequence:

[tex]8,\;2,\;\dfrac{1}{2},\;\dfrac{1}{8},\;\dfrac{1}{32}[/tex]

The common ratio for a geometric sequence can be found by dividing a term by the previous term.

[tex]\implies r=\dfrac{a_2}{a_1}=\dfrac{2}{8}=\dfrac{2 \div 2}{8 \div 2} =\dfrac{1}{4}[/tex]

Therefore, the common ratio for the given geometric sequence is 1/4.

(A) There was a change from 1995 and 2003 at a rate of -0.1463

(B) A change took place between 1995 and 2003 at a rate of -14.63% per year.

(C) In 2007, the car would be worth $5,844.24.

The ability to reclaim the purchase price or other basis of a specific item over the period of its usage is provided through depreciation, an annual income tax deduction.

It is an allowance for the regular deterioration, wear, tear, or obsolescence of the asset.

The worth of the car decreases over time.

Depreciation is the term for this action.

The value of an asset decreases over time as a result of depreciation, or ordinary wear and tear.

To determine the annual rate of change, use the following formula:

g = (FV/PV)¹⁾ⁿ - 1

Now, compute the following value using the formula:

(11000/3900)¹⁾⁸ - 1

-0.1463 = -14.63%

To calculate a car's value in 7 years, apply the formula below:

FV = P (1 + g)ⁿ

$39,000 x (1 - 0.1463)¹²

$39,000 x 0.8537¹² = $5,844.24

Therefore, (A) There was a change between 1995 and 2003 at a rate of -0.1463

(B) A change took place between 1995 and 2003 at a rate of -14.63% per year.

(C) In 2007, the car would be worth $5,844.24.

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Correct question:

A car was valued at $39,000 in the year 1995. The value depreciated to $11,000 by the year 2003.

A)What was the annual rate of change between 1995 and 2003? (Round to 4 decimal places)

B)What is the correct answer to part A written in percentage form?

C)Assume that the car value continues to drop by the same percentage. What will the value be in the year 2007?

Based on the given information, the measure of angle Y is 30 degrees.

A transformation is a process or operation that changes the position, shape, or orientation of a geometric figure or an object in space.

There are several types of transformations, including:

Rotation: A transformation that rotates a figure around a fixed point, called the center of rotation, by a certain angle.

Reflection: A transformation that produces a mirror image of a figure across a line, called the line of reflection.

To start, let's first find the vertices of triangle XYZ after the given transformations:

Rotation of 90 degrees counterclockwise about the origin:

The rotation of point (x, y) by 90 degrees counterclockwise about the origin gives us the point (-y, x). So we can apply this transformation to each vertex of triangle ABC to get the vertices of triangle XYZ after rotation:

A = (cos(45), sin(45)) = (√2/2, √2/2) → A' = (-√2/2, √2/2)

B = (cos(55), sin(55)) → B' = (-sin(55), cos(55))

C = (cos(80), sin(80)) → C' = (-sin(80), cos(80))

Reflection across the line y=x:

To reflect a point (x, y) across the line y=x, we simply swap the x and y coordinates to get (y, x). So we can apply this transformation to each vertex of triangle XYZ to get the final vertices:

A'' = (√2/2, -√2/2)

B'' = (cos(55), -sin(55))

C'' = (cos(80), -sin(80))

Now we can use the law of cosines to find the measure of angle Y:

cos(Y) = [B''C''² + A''C''² - A''B''²] / [2 * B''C'' * A''C'']

= [cos(55)² + cos(80)² - (√2/2)²] / [2 * cos(55) * cos(80)]

≈ 0.866

So we have cos(Y) = 0.866, which means Y ≈ 30 degrees. Therefore, the measure of angle Y is approximately 30 degrees.

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The answer is (a) 203.5 bacteria, rounded to the nearest whole number.

Percentage is a way of expressing a number or proportion as a fraction of 100. It is represented by the symbol "%".

The growth rate of E.Coli is given as 1.9% per minute, which can be expressed as 0.019 per minute as a decimal.

Let N(t) be the number of E.Coli present at time t. Then, we can write the differential equation governing the population growth as:

dN/dt = rN,

where r is the growth rate (0.019 per minute in this case).

The general solution to this differential equation is:

N(t) = [tex]N0e^{(rt)}[/tex]),

where N0 is the initial population size.

We are given that the current number of E.Coli present is 172, so N0 = 172. We want to find N(8.5), so we plug in t = 8.5 and solve:

N(8.5) = [tex]172e^{0.019*8.5}[/tex] ≈ 203.5

Therefore, the answer is (a) 203.5 bacteria, rounded to the nearest whole

number.

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Answer:

(x, y) → (-x, y)

Step-by-step explanation:

The coordinates of the points that make up the function are equal to (x, y).

If we reflect the function over the y-axis, the sign of the y-value stays the same due to still being on the same side of the x-axis. The x-value sign will be the opposite of its current value due to being reflected over the y-axis.

This leads us to the algebraic expression:

(x, y) → (-x, y)

The estimated yield of wheat when the average amount of fertilizer is applied is 68.65 units.

To estimate the yield of wheat when the average amount of fertilizer is applied, we first need to calculate the average amount of fertilizer from the given data.

Use a linear regression tool to fit a line to the data points.

[tex]y = mx + b[/tex]

where y is the yield of wheat, x is the amount of fertilizer, m is the slope of the line, and b is the y-intercept.

Using this method, we can estimate the yield of wheat when no fertilizer is applied.

To estimate the yield of wheat when the average amount of fertilizer is applied, we first need to calculate the average amount of fertilizer from the given data.

[tex](50 + 80 + 100 + 122 + 150 + 180 + 200) / 7 = 120[/tex]

Therefore, the average amount of fertilizer applied is 120.

[tex]y = m(120) + b[/tex]

To get the values of m and b, we can perform a linear regression analysis on the given data points.

[tex]m = 0.3903[/tex]

b = 21.754

Substituting these values into the equation above, we get:

[tex]y = 0.3903(120) + 21.754 = 68.65[/tex]

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If 88% of the stamps were foreign, then the remaining 12% must be domestic stamps.

To find out how many domestic stamps Blake collected, we can start by calculating what 12% of 275 is:

12% of 275 = 0.12 x 275 = 33

The remaining stamps are foreign stamps.

Therefore, Blake collected 33 domestic stamps.

Answer:

P = 3

Step-by-step explanation:

[tex] {p}^{3} = 27[/tex]

[tex]p = \sqrt[3]{27} = 3[/tex]

Answer:

P = 3

Step-by-step explanation:

Write down the equation:

p^3 = 27

Then divide both sides by cubed:

p = 3

The maximum value of Q was found to be 26.49 subject to the cost constraint K+4L=16.2, and the optimal value of Q decreased by approximately 9.39 when the cost constraint was changed to K+4L=17.

To find the maximum value of Q subject to the cost constraint, we can use the method of Lagrange multipliers. We first define the Lagrangian function:

L(K, L, λ) = 10√(KL) + λ(K + 4L - 16.2)

where λ is the Lagrange multiplier.

∂L/∂K = 5√(L/K) + λ = 0

∂L/∂L = 5√(K/L) + 4λ = 0

∂L/∂λ = K + 4L - 16.2 = 0

K = 3.24, L = 2.16, λ = -2.5

Therefore, the maximum value of Q is:

Q = 10√(3.24*2.16) = 10√7.0224 ≈ 26.49

If the cost constraint is changed to K+4L=17, we can repeat the same process with the new constraint to get:

K = 4.25, L = 0.6875, λ = -2.5

Therefore, the new maximum value of Q is:

Q = 10√(4.25*0.6875) = 10√2.9297 ≈ 17.10

The change in the optimal value of Q is then:

17.10 - 26.49 ≈ -9.39

The optimal value of Q decreases by approximately 9.39 when the cost constraint is changed from K+4L=16.2 to K+4L=17. This indicates that the optimal value of Q is sensitive to changes in the cost constraint.

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Find the maximum value of Q = 10√/KL subject to the cost constraint K+4L=16.

Estimate the change in the optimal value of Q if the cost constraint is changed to K+4L =17. Interpret the meaning of Lagrange multiplier.

the volume of one scoop of ice cream is 523.6 cc. The cup can hold a volume of 942.5 cc, which is more than one scoop of ice cream. Therefore, Moses's cup can hold one scoop of ice cream without overflowing or spilling.

let's break down the problem step by step.

First, we need to calculate the volume of one scoop of ice cream. Since each scoop is a perfect sphere with a radius of 5 cm, we can use the formula for the volume of a sphere, which is:

V = (4/3)πr³

where V is the volume and r are the radius. Substituting r = 5 cm, we get:

V = (4/3)π(5³) = 523.6 cubic centimeters (cc)

So, the volume of one scoop of ice cream is 523.6 cc.

Next, we need to calculate the volume of the cup and the cone. Since both have the same radius as the scoop of ice cream, we can use the formula for the volume of a cylinder and a cone, respectively, with height 12 cm and radius 5 cm:

For the cup:

[tex]V_{cup}[/tex]= πr²h = π(5²) (12) = 942.5 cc

For the cone:

[tex]V_{cone}[/tex] = (1/3)πr²h = (1/3)π(5²)(12) = 314.2 cc

So, the volume of the cup is 942.5 cc, and the volume of the cone is 314.2 cc.

Now, we can determine how much ice cream each container can hold. Since each person gets one scoop of ice cream, we need to compare the volume of one scoop of ice cream to the volume of each container.

For Gaby's cone:

We know that the volume of one scoop of ice cream is 523.6 cc. The cone can hold a volume of 314.2 cc, which is less than one scoop of ice cream. Therefore, Gaby's cone can only hold one scoop of ice cream.

For Moses's cup:

Again, the volume of one scoop of ice cream is 523.6 cc. The cup can hold a volume of 942.5 cc, which is more than one scoop of ice cream. Therefore, Moses's cup can hold one scoop of ice cream without overflowing or spilling.

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The complete question: What is the volume of ice cream in each scoop and what is the total volume of ice cream that Mr. Woods needs to buy for Moses and Gaby? Also, what is the total surface area of the ice cream exposed to the air in each container (cone and cup)?

Answer:

stretch by 2, right 4, down 1

Step-by-step explanation:

In a function in the form:

f(x) = a(x - h)² + k

the number in front, a, shows vertical stretch (if its bigger than 1. Its a shrink or smash if its a fraction less than 1)

Horizontal shifts are shown in close next to the x, inside the parenthesis shown here with an h. Horizontals shifts are maybe the opposite of what you might think they should be. + anumber is a left shift and - anumber is a right shift.

Vertical shifts are shown by a number tacked onto the end of the equation, k These shifts are more intuitive. + anumber shifts up and - anumber shifts down.

For your equation:

f(x) = 2(x-4)²-1

The 2 is a vertical stretch. The -4 by the x is a right shift 4. The -1 tacked into the end is a shift down 1

The rectangular prism, which has sides measuring 12 feet, 14 feet, and 20 feet, has a surface area of 1376feet².

The space a two-dimensional flat surface occupies is the area. Its unit of measurement is the square. A three-dimensional object's surface area is the area occupied by its exterior surface. It is also measured in units².

We must first determine the surface area of each face of a rectangular prism with sides measuring 12 feet, 14 feet, and 20 feet before adding them all up.

Let's first determine the area of the top and bottom faces, which are both rectangles measuring 12 feet by 20 feet each:

480feet² is equal to 2 x (12 ft x 20 ft) for the top and bottom faces.

The front and back faces, which are both rectangles with measurements of 12 feet by 14 feet, can now be calculated as follows:

2 x (12 ft) x (12 ft) x (12 ft) = 336feet² is the area of the front and back faces.

Let's finally determine the area of the rectangles that make up the left and right faces, both of which measure 14 feet by 20 feet:

2 x (14 ft x 20 ft) = 560feet² is the area of the left and right faces.

We add up the areas of all six faces to determine the total surface area:

Overall surface area equals the sum of the areas on the top, bottom, front, and back, as well as the left and right faces.

Overall surface area equals 480feet² + 336feet² + 560feet² =1376 feet².

As a result, the rectangular prism with measurements of 12 feet, 14 feet, and 20 feet has a surface area of 1376 feet².

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The complete question is:

what is the surface area of 12 ft 16 ft 14 ft 20 ft

Answer: $4845

Step-by-step explanation:

First, round Peter's weekly payment to the nearest whole dollar:

$56.74 ≈ $57

Next, multiply the rounded weekly payment by the number of weeks he needs to make payments:

$57 * 85 weeks = $4845

So, Peter will pay back an estimated total of $4845.

Credit card issuers use credit scores to determine the consumer's eligibility for credit and to set pricing for credit lines. (Options C and E).

A credit card is a type of payment card that allows the cardholder to borrow money from a financial institution or issuer to purchase goods or services. The cardholder can use the card to make purchases up to a certain credit limit set by the issuer, and they are required to pay back the borrowed amount, usually with interest, within a specified period of time.

Credit cards typically offer a range of benefits, such as rewards programs, cash back, and fraud protection, and they are widely used for online and in-person transactions. However, if not used responsibly, credit cards can lead to debt and financial problems.

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Answer:

Step-by-step explanation:

To simplify 5(y+2) +5(y-1), we distribute the 5 to each term inside the parentheses, and then combine like terms as follows:

5(y+2) +5(y-1)

= 5y + 10 + 5y - 5 // Distribute the 5

= 10y + 5 // Combine like terms

Therefore, 5(y+2) +5(y-1) simplifies to 10y + 5.

Answer:

To find the distance between two points, we use the distance formula: distance = square root of [(x2 - x1)^2 + (y2 - y1)^2] In this case, our points are (4, -3) and (-7, -3), so we can plug in the values: distance = square root of [(-7 - 4)^2 + (-3 - (-3))^2] distance = square root of [(-11)^2 + (0)^2] distance = square root of (121) distance = 11 Therefore, the distance between the points (4, -3) and (-7, -3) is 11 units.

The linear equation that represents the cost of visiting the gym x times is: y = 2x + 25

What is Algebraic expression ?

An algebraic expression is a mathematical phrase that can contain variables, constants, and operators (such as addition, subtraction, multiplication, and division) that are used to represent quantities and their relationships.

The cost of joining a gym includes an initial fee of $25 and then $2 per visit. Let y be the total cost of visiting the gym x times.

To find the equation that represents this relationship, we need to determine the slope and the y-intercept.

The slope represents the rate of change of the cost with respect to the number of visits, which is the cost per visit. Since the cost per visit is $2, the slope is 2.

The y-intercept represents the initial cost of joining the gym, which is $25.

Therefore, the linear equation that represents the cost of visiting the gym x times is: y = 2x + 25

In this equation, x represents the number of visits to the gym, and y represents the total cost of those visits.

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Therefore , the solution of the given problem of unitary method comes out to be  1,440 students who favor ice cream

The task may be completed using this generally accepted ease, existing variables, and any important components from the original Diocesan customizable query. If so, you might get another opportunity to work with the object. Otherwise, algorithmic evidence will no longer be affected by any significant factors.  

Here,

The table shows that 81 of the 225 students favor ice cream.

Out of the 4,000 students, we can use the following proportion to estimate how many favor ice cream:

=> 81/225 = x/4000

If we cross-multiply, we obtain:

=> 225x = 81 * 4000

If we simplify, we get:

=> 225x = 324,000

By multiplying both parts by 225, we obtain:

=> x = 1440

The college will have about 1,440 students who favor ice cream, so that is the best guess as to how many scoops of ice cream the college will require.

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Answer:

c

Step-by-step explanation:

To find the x-intercept, we set y to 0 and solve for x:

0 = x + 8

x = -8

Therefore, the x-intercept is -8.

To find the y-intercept, we set x to 0 and solve for y:

y = 0 + 8

y = 8

Therefore, the y-intercept is 8.

Hopes this helps

Answer:

AB = 9

Step-by-step explanation:

Since triangle ACE and BCD are congruent, they have the same angles, you can figure out the fraction to get from ACE to BCD, then reverse it. So since CE is 12 + 4 = 16 and CD is 4, we can see that ACE is 4 times greater than BCD because 16/4 = 4.

So to get AB, find AC by multiplying BC by 4, which gives 12, then subtract AC by BC to get AB: 12 - 3 = 9