please help rewarding brainliest

Answer:

its better to buy the cylinder because it can hold more

Step-by-step explanation:

Without specific dimensions for the containers, we cannot provide exact values for volume and unit rate.

To determine which popcorn container has a better unit rate and holds more popcorn, we first need to find the volume of both containers.

Assuming the given dimensions are radius and height, let's say the cone has a radius of 'r1' and a height of 'h1,' and the cylinder has a radius of 'r2' and a height of 'h2.'

The volume of a cone (V1) is given by the formula: V1 = (1/3)πr1²h1
The volume of a cylinder (V2) is given by the formula: V2 = πr2²h2

Next, we need to find the unit rate of both containers. Divide the volume of each container by its price.

Unit rate of cone: UR1 = V1 / $6.75
Unit rate of cylinder: UR2 = V2 / $6.25

To determine which container holds the most popcorn, compare V1 and V2. The container with the larger volume holds more popcorn.

To determine which container has a better unit rate, compare UR1 and UR2. The container with the higher unit rate is a better buy per $1.

However, you can plug in the given dimensions for your problem and follow these steps to find your answer.

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Thus, the number of coin for each are- number of dimes be 5 and number of quarters be 1.

A linear system can be algebraically solved using the substitution approach. One y-value is substituted for another in the substitution procedure. Getting the value of the x-variable in regards of the y-variable is the method's most straightforward step.

Let the number of dimes be 'x'.

Let the number of quarters be 'y'.

Then,

Total amount of parking meter is 6.25.

x + y = 6.25

x = 6.25  - y ....eq 1

Now,

number of dimes is 2 more than 3 times the number of quarters.

So,

x = 3y + 2  ...eq 2

Equating eq 1 and 2

6.25  - y = 3y + 2

4y = 4.25

y = 1.062

y = 1 (approx)

x = 3(1.065) + 2

x = 5.195

x = 5 (approx)

Thus, the number of coin for each are- number of dimes be 5 and number of quarters be 1.

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An equation represents how much will be in the account after 6 years is M = 9,100*[tex](1 + 0.05/1)^{1*6}[/tex]

What is an account ?

The formula to calculate the amount in a savings account after a certain number of years with no deposits or withdrawals is:

M = P*[tex](1 + r/n)^{n*t}[/tex]

Where:

P = Principal (initial amount)

r = Annual interest rate (as a decimal)

n = Number of times the interest is compounded per year

t = Time in years

In this case, P = $9,100, r = 0.05 (5% as a decimal), n = 1 (interest is compounded annually), and t = 6 (years).

Using these values, we can calculate the amount in the account after 6 years as follows:

M = 9,100*[tex](1 + 0.05/1)^{1*6}[/tex]

M = 9,100*[tex](1.05)^{6}[/tex]

M = 9,100(1.3401)

M = $12,175.91

Therefore, the correct equation is:

M = 9,100*[tex](1 + 0.05/1)^{1*6}[/tex]

What is an interest?

Interest is the cost of borrowing money, usually expressed as a percentage of the amount borrowed, and is paid by the borrower to the lender. It can also be the amount earned on an investment or savings account.

Interest rates can vary depending on a variety of factors, including the current economic conditions, inflation, the borrower's creditworthiness, and the length of the loan or investment term.

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Complete question is: 9,100 dollars is placed in a savings account with an annual interest rate of 5%. If no money is added or removed from the account. An equation  M = 9,100*[tex](1 + 0.05/1)^{1*6}[/tex] represents how much will be in the account after 6 years.

Answer:

it’s the first one- 9x-3=60

Step-by-step explanation:

Answer: 16a^2−24a−7

Step-by-step explanation:

Ike and Gus ran for 12 miles and 36 miles respectively.

Let us suppose, the number of miles Ike ran be x.

According to question, Gus ran 3 times as far as Ike.

So, Gus's running distance = 3x.

Also, Gus's and Ike's combined running distance = 48

⇒ x + 3x = 48

⇒ 4x = 48

⇒ x = 48 ÷ 4

⇒ x = 12.

∴ Ike ran 12 miles.

and distance covered by Gus = 3x

                                                 = 3 × 12

                                                 = 36.

Hence, Ike and Gus ran for 12 miles and 36 miles respectively.

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A pre-image is a transformation representing a flip of a figure. For reflection:

The preimage and the image have the same size.

An image created by a reflection will always be congruent to its pre-image.

Corresponding angles and segments are always congruent in a reflection of a figure.

An image and its pre-image are always the same distance from the line of reflection.

Transformation

Transformation is the movement of a point from its initial location to a new location. Types of transformation are rotation, translation, reflection, and dilation.

A reflection is a transformation representing a flip of a figure. Reflection preserves the shape and size of the figure. Also, An image and its pre-image are always the same distance from the line of reflection.

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Answer:

Greatest is Miami

Least is Washington

Step-by-step explanation:

Population/Area= Population Density

Ex:

Population 2

Area 1

2/1=2

Based on the given information and the secant-Secant Theorem, we have derived the equation (XY * XZ) = (10X * 10X).

Given that Circle U has points X and Z on the circle, and secant segments XY and ZY intersect at point Y outside the circle, we will use the following terms in our answer: Circle, Secant Segments, Intersection, and Ratio.

Step 1: Identify the segments involved.
We have two secant segments intersecting at point Y: XY and ZY.

Step 2: Apply the secant-secant theorem.
According to the secant-secant theorem, the product of the lengths of the segments from the intersection point to the points on the circle should be equal. In other words, (XY * XZ) = (ZY * ZY).

Step 3: Apply the given ratio.
We are given that ZY = 10X. Let's substitute this into the equation from Step 2: (XY * XZ) = (10X * 10X).

Step 4: Solve for XY and XZ.
Now, we need to solve for XY and XZ using the equation from Step 3. However, since we only have one equation and two variables, we cannot solve it completely. We need more information to find the specific values of XY and XZ.

Based on the given information and the secant-secant theorem, we have derived the equation (XY * XZ) = (10X * 10X). To find the specific values of XY and XZ, additional information is required.

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Answer:

First problem: x = √15

Second problem: x = 6√5 (B)

Third problem: x = 6, y = 12 (A)

Fourth problem: x = 6√2 (C)

Step-by-step explanation:

We use geometric means as follows:

First problem:

[tex] \frac{3}{x} = \frac{x}{5} [/tex]

[tex] {x}^{2} = 15[/tex]

[tex]x = \sqrt{15} [/tex]

Second problem:

[tex] \frac{20}{x} = \frac{x}{9} [/tex]

[tex] {x}^{2} = 180[/tex]

[tex]x = \sqrt{180} = \sqrt{36} \sqrt{5} = 6 \sqrt{5} [/tex]

So B is the correct answer.

Third problem:

We have a 30°-60°-90° triangle.

x = 6, y = 12. So A is the correct answer.

Fourth problem:

We have a right isosceles triangle.

x = 12/√2 = 6√2. So C is the correct answer.

The increasing levels of carbon dioxide in the atmosphere are causing significant changes to the hydrosphere, with far-reaching impacts on both marine and freshwater ecosystems.

What is Carbon dioxide ?

Carbon dioxide (CO2) is a gas that is naturally present in the Earth's atmosphere, and it plays a vital role in regulating the Earth's climate.

The increasing levels of carbon dioxide in the atmosphere are having a significant impact on the hydrosphere, which includes all the Earth's water systems such as oceans, lakes, rivers, groundwater, and ice caps.

When carbon dioxide dissolves in water, it reacts with it to form carbonic acid, which can cause the pH of the water to decrease, making it more acidic. This process is known as ocean acidification. As carbon dioxide levels continue to rise, the acidity of seawater is also increasing, which can have negative impacts on marine organisms that depend on certain pH levels to survive. For example, ocean acidification can interfere with the ability of shell-forming organisms such as corals, mollusks, and some plankton to build their shells, which can lead to population declines and changes in the food web.

In addition to ocean acidification, increased levels of carbon dioxide in the atmosphere can also contribute to rising sea levels due to melting ice caps and glaciers. The warming caused by increased greenhouse gases can also cause changes in precipitation patterns, leading to more frequent and severe floods and droughts that can impact freshwater resources.

Therefore,  the increasing levels of carbon dioxide in the atmosphere are causing significant changes to the hydrosphere, with far-reaching impacts on both marine and freshwater ecosystems.

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[tex]Therefore, when[/tex] [tex]x=4,y=1.125[/tex].  Y varies inversely with the cube of x and y=9 when x=2.

A cube is a three-dimensional geometric shape that has six square faces, all of which are congruent to each other, and eight vertices (corners) where three faces meet. It is a regular polyhedron, which means all its faces are congruent and all its angles are equal.

A cube is often used in mathematics, geometry, and engineering because of its regular shape, which makes it easy to calculate and work with. It is also a common shape in everyday objects, such as dice, Rubik's Cubes, and boxes. The volume of a cube is calculated by multiplying the length of any of its sides by itself twice, while the surface area is found by multiplying the length of any of its sides by six.

If y varies inversely with the cube of x, we can write this relationship as:

[tex]y = k/x^3[/tex]

where k is a constant of proportionality. To solve for k, we can use the given information that y=9 when x=2:

[tex]9 = k/2^3[/tex]

Simplifying the equation, we get:

[tex]k = 9 * 2^3[/tex]

[tex]k = 72[/tex]

Now that we have determined the value of k, we can use the equation to find the value of y for any value of x. For example, if x=4, we have:

[tex]y = 72/4^3[/tex]

[tex]y = 72/64[/tex]

[tex]y = 1.125[/tex]

[tex]Therefore, when=4,y=1.125[/tex]

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The perimeter of the trapezoid is 16 units

The formula used for calculating the perimeter of a trapezoid is expressed with the equation;

P = a + b + c + d

Such that the parameters of the equation are;

We then have;

a = 1 units

b= 9 units

c = 3 units

d = 3

Substitute the values

Perimeter = 1 + 9 + 3 + 3

Add the values

Perimeter = 16 units

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1331.046

Step-by-step explanation:

Quadratic function that models the given data points: [tex]y &= -7.5x^2 + 13.5x + 42[/tex].

How to write a function that models the data?

To model the data shown in the graph, we need to find a function that best fits the given data points. A polynomial is a common type of function used for this purpose. In this case, we can use a second-degree polynomial (a quadratic function) to fit the data.

[tex]y &= ax^2 + bx + c[/tex]

where a, b, and c are constants governing the shape and position of the parabola.

To find the values of a, b, and c that fit the given data points, we can use a system of equations. We can substitute the x and y values of each point into the equation of the quadratic function and get three equations:

[tex]a(0)^2 + b(0) + c &= 42 \\a(1)^2 + b(1) + c &= 21 \\a(2)^2 + b(2) + c &= 10.5 \\[/tex]

Simplifying these equations, we get:

c = 42

a + b + c = 21

4a + 2b + c = 10.5

Substituting c = 42 into the second equation, we get:

a + b = -21

Substituting c = 42 into the third equation and simplifying, we get:

4a + 2b = -73.5

Solving these two equations simultaneously, we get:

a = -7.5

b = 13.5

Substituting these values into the equation of the quadratic function, we get:

[tex]y = -7.5x^2 + 13.5x + 42[/tex]

This function models the data shown in the graph. We can verify this by plotting the function and the data points on the same graph and checking that they match.

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The integral of  probability density function [tex]\int_{4}^{\infty}[/tex]f(x) dx represents option b. the probability that a tree has a diameter of at least 4 inches.

Probability density function represented by function f.

Probability density function f for the diameter of trees in a forest is equals to ,

[tex]\int_{4}^{\infty}[/tex]f(x) dx

Because the integral is computing the area under the PDF curve for diameters greater than or equal to 4 inches.

And the area under a PDF curve represents the probability of the random variable in this case, tree diameter falling within that range.

This implies,

Integrating the PDF from 4 to infinity gives the probability of a tree having a diameter greater than or equal to 4 inches.

Therefore, the correct answer to represents the probability density function is Option (b). the probability that a tree has a diameter of at least 4 inches.

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The above question is incomplete , the complete question is :

Let f be the probability density function (PDF) for the diameter of trees in a forest, measured in inches. What does [tex]\int_{4}^{\infty}[/tex] f(x) dx represent?

(a) The standard deviation of the diameter of the trees in the forest.

(b) The probability that a tree has a diameter of at least 4 inches

(c) The probability that a tree has diameter less than 4 inches.

(d) The mean diameter of the trees in the forest.

Answer:

99°

Step-by-step explanation:

180° - 36° - 63° = 81

360° - 180° - 81° = 99°

Charlie must earn at least 125 tickets in the last game tο win the big prize.

A linear inequality is a mathematical statement that compares twο expressiοns using the symbοls < (less than), > (greater than), ≤ (less than οr equal tο), οr ≥ (greater than οr equal tο), and where bοth expressiοns are linear functiοns οf the same variables. It describes range οf values that the variables can take while satisfying the inequality.

Let x be the number οf tickets that Charlie must earn in the last game tο win the big prize.

The tοtal number οf tickets Charlie has won so far is the sum οf the tickets frοm all the games he has played:

95 + 115 + 90 + 75 = 375.

Tο win the big prize, Charlie must earn at least 500 tickets in tοtal.

Therefοre, we can write the fοllοwing inequality tο sοlve fοr the pοssible number οf tickets, x:

375 + x ≥ 500

Simplifying the inequality:

x ≥ 500 - 375

x ≥ 125

Therefore, Charlie must earn at least 125 tickets in the last game to win the big prize.

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Answer:

[tex](b - c)(b + c - 10)[/tex]

Step-by-step explanation:

[tex] {b}^{2} - {c}^{2} - 10(b - c)[/tex]

What does factorising mean?

What does expanding brackets mean?

Now, expand the brackets in this expression:

[tex] {b}^{2} - {c}^{2} - 10b + 10c[/tex]

Apply the difference of squares formula to factor the expression even more (also, factor out -10 from the expression by putting it in front of the brackets):

[tex](b - c)(b + c) - 10(b - c)[/tex]

Now, factor out (b - c) from the expression:

[tex](b - c)(b + c - 10)[/tex]

the possible whole number measures of the third side are 5, 6, and 7, listed in ascending order.

what is ascending order?

Ascending order refers to a sorting arrangement in which items are arranged from smallest to largest or from lowest to highest. For example, if you have a list of numbers such as 2, 7, 1, 9, 5, arranging them in ascending order would give you the sequence

In the given question,

According to the Triangle Inequality Theorem, the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. Therefore, in this case, we have:

6 + 2 > x

Where x is the length of the third side. Simplifying this inequality, we get:

8 > x

So, the third side must be less than 8.

Now, we also know that the third side must be greater than the difference between the first two sides:

6 - 2 < x

4 < x

So, the third side must be greater than 4.

Therefore, the possible whole number measures of the third side are 5, 6, and 7, listed in ascending order.

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From midnight to 5:00 am, the temperature dropped for 5 hours at 0.5°C each hour, so the total temperature drop was:

5 x 0.5°C = 2.5°C

Let the temperature at midnight be T°C. Then, we can use the information given to set up an equation:

Temperature at 5:00 am = Temperature at midnight - total temperature drop

11.5°C = T°C - 2.5°C

Solving for T, we can add 2.5°C to both sides of the equation:

11.5°C + 2.5°C = T°C - 2.5°C + 2.5°C

14°C = T°C

Therefore, the temperature at midnight was 14°C.

The confidence interval with the smaller margin of error will be the one with the smaller sample proportion difference.

(a) To calculate the point estimate of the proportion of respondents who think the country is headed in the right direction, divide the number of "yes" responses by the total number of respondents (2,500).
Point estimate = (number of "yes" responses) / 2,500
(b) To find the margin of error at 95% confidence, use the formula:
Margin of error = Z * √(p(1-p) / n)
Here, Z = 1.96 (from the standard normal distribution for 95% confidence), p is the point estimate calculated in part (a), and n = 2,500.
(c) To find the 95% confidence interval for the proportion of respondents who think the country is headed in the right direction, use the formula:
Confidence interval = point estimate ± margin of error
(d) To find the 95% confidence interval for the proportion of respondents who do not think the country is headed in the right direction, first calculate the point estimate for "no" responses:
Point estimate (no) = (number of "no" responses) / 2,500
Then calculate the margin of error and confidence interval as in parts (b) and (c).
(e) To determine which confidence interval has a smaller margin of error, compare the margin of error calculated in parts (b) and (d).

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The graph of the linear equation y = mx + c is a line with m as the slope, and m, and c as the y-intercept. This form of the linear equation is called the slope-intercept form, and the values of m and c are real numbers.

The slope, m, represents the steepness of a line. The slope of the line is also termed a gradient, sometimes. The y-intercept, b, of a line, represents the y-coordinate of the point where the graph of the line intersects the y-axis.

Here, the distance c is called the y-intercept of the given line L.

So, the coordinate of a point where the line L meets the y-axis will be

(0, c). That means line L passes through a fixed point (0, c) with slope m.

We know that, the equation of a line in point-slope form, where (x1, y1) is the point and slope m is:

(y – y1) = m(x – x1)

Here, (x1, y1) = (0, c)

Substituting these values, we get;

y – c = m(x – 0)

y – c = mx

y = mx + c

Therefore, the point (x, y) on the line with slope m and y-intercept c lies on the line if and only if y = mx + c

Note: The value of c can be positive or negative based on the intercept made on the positive or negative side of the y-axis, respectively

Slope Intercept Form x Intercept

We can write the formula for the slope-intercept form of the equation of line L whose slope is m and x-intercept d as:

y = m(x – d)

Here,

m = Slope of the line

d = x-intercept of the line

Sometimes, the slope of a line may be expressed in terms of tangent angle such as:

m = tan θ

Zach's hourly wage is determined by taking the basic rate of $16.00 and adding $2.00 for each extra student beyond the first one in the calculation is P = $16.00 + ($2.00 x (n - 1))

That's an interesting situation! It seems like Zach's hourly pay for tutoring is not fixed, but rather increases with each additional student he tutors. Here's how his pay works:

If Zach is tutoring just one student, he gets paid $16.00 per hour.

If a second student joins the tutoring session, Zach's hourly pay increases to $18.00 per hour ($16.00 for the first student plus $2.00 for the second).

If a third student joins, Zach's hourly pay increases to $20.00 per hour ($16.00 for the first student plus $2.00 for the second and third students).

And so on, for each additional student who joins the tutoring session.

So Zach's hourly pay is not a fixed rate, but rather a function of the number of students he is tutoring.

If we let P be Zach's hourly pay and n be the number of students he is tutoring, we can express Zach's pay as:

P = $16.00 + ($2.00 x (n - 1))

This formula calculates Zach's hourly pay by taking the base rate of $16.00 and adding $2.00 for each additional student beyond the first one.

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Question:-  Zach got hired to tutor math. He gets paid $16.00 per hour for tutoring one student. Each student that joins the tutoring session increases his hourly pay by $2.00.

The total number of ice cream which is in the cartons if cartons to get one liter (L) of ice cream is 480 Liters.

The fundamental concept of repeatedly adding the same number is represented by the process of multiplication. The results of multiplying two or more numbers are known as the product of those numbers, and the components that are multiplied are referred to as the factors. Repeated addition of the same number is made easier by multiplying the numbers.

Mathematicians use multiplication to calculate the product of two or more integers. It is a fundamental operation in mathematics that is frequently utilised in everyday life. When we need to combine groups of similar sizes, we utilise multiplication.

We have,

throw out the 13 least popular flavors,

Total flavors

25 - 13 = 12

80/2 = 40 liters.

The remaining 12 flavors and 40 liters each so,

total ice cream = 12 x 40 = 480 liters.

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Complete question:

An ice cream shop is testing some new flavors of ice cream. They invent 25 new flavors for customers to try. and throw out the 13 least popular flavors. The shop makes 80 cartons of each of the remaining flavors. It takes cartons to get one liter (L) of ice cream​. How much total ice cream is in the cartons?

We may assume that out of the 300 graduating students, 120 plan to go

to college, 75 want to work, 60 are unsure, and 45 want to join the

military. It's critical to remember that these are estimates based on a poll

of 20 peers, and actual figures may vary slightly.

Based on the data from the survey of 20 classmates, we can make

inferences about the plans of the entire graduating class of 300 students

using proportions.

Let's fill in the table based on the given information:

Plan Number of classmates Proportion

College 8 8/20 = 0.4

Work 5 5/20 = 0.25

Undecided 4 4/20 = 0.2

Military 3 3/20 = 0.15

To find the number of students in each category out of 300, we can

multiply the proportion by 300:

College: 0.4 x 300 = 120 students

Work: 0.25 x 300 = 75 students

Undecided: 0.2 x 300 = 60 students

Military: 0.15 x 300 = 45 students

Therefore, we can infer that out of the 300 students in the graduating

class, approximately 120 plan to attend college, 75 plan to work, 60 are

undecided, and 45 plan to enter the military. It's important to note that

these are estimates based on the survey of 20 classmates, and there

may be some variability in the actual numbers.

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The equation of the exponential function is[tex]4 * 3^{x}[/tex].

What is exponential function?

An exponential function is  of the form:

[tex]f(x) = a^{x}[/tex]

here "a" is a constant called the base, and "x" is the variable. The base "a" is typically a positive real number, but it can also be a negative number or a complex number. The function is called exponential because the variable "x" appears in the exponent.

We can see that as "x" increases by 1, "y" is multiplied by a constant factor of 3. This suggests that the exponential function has a base of 3. To find the equation, we can use the general form of an exponential function:

[tex]y = a * b^{x}[/tex]

where "a" is the initial value or y-intercept, and "b" is the base.

We can use the given values of (x,y) to form a system of equations:

[tex]a * 3^{0} = 4[/tex]

[tex]a * 3^{1} = 12[/tex]

[tex]a * 3^{2} = 36[/tex]

[tex]a * 3^{3} = 108[/tex]

Simplifying each equation, we get:

a = 4

3a = 12

9a = 36

27a = 108

Solving for "a", we get:

a = 4

Substituting this value of "a" into the equation, we get:

[tex]y = 4 * 3^{x}[/tex]

So the equation of the exponential function represented by the table is y [tex]= 4 * 3^{x}.[/tex]

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Answer:

y = -x² - 2x + 15

Step-by-step explanation:

We can see that upto the x-value of -1, the y-value increases, and after -1, the y-value decreases. This tells us both that (-1, 16) is the vertex and that the function opens downwards.

The typical vertex form for quadratic equations is:

y= a(x-h)² + k where (h,k) is the vertex.

Replace the vertex (-1,16) for (h,k)

y = a (x+1)² +16

To find a, replace any value from the table. Let's use (3,0).

0 = a (3+1)² +16

0 = a (4)² + 16

0 = 16a + 16

-16 = 16a

a = -1

Now, insert that a into our  equation...

y = -13/16 (x+1)² +16

Then simplify:

y = -1 (x² +2x +1) + 16

y = -x² -2x -1 +16

We get our answer:

y = -x² - 2x + 15

Answer:

  50 ft by 25 ft . . . . . 50 ft parallel to the barn

Step-by-step explanation:

You want the dimensions of the largest rectangular area that can be enclosed using 100 ft of fence for three sides.

If the dimensions of the space are L feet in length and W feet in width, where L is parallel to the barn, the length of the perimeter fence is ...

  P = L +2W

Solving for W gives ...

  W = (P -L)/2

The area of the enclosed space is ...

  A = LW

  A = L(P -L)/2 . . . . . substitute for W

The area formula is the equation for a parabola that opens downward. It has zeros at L=0 and at L=P. The vertex (maximum) is found at the value of L that lies on the line of symmetry, halfway between these zeros. If we call that length M, then we have ...

  M = (0 +P)/2 = P/2

The length of enclosure that maximizes the area is 1/2 the length of the available fence.

The width is ...

  W = (P -P/2)/2 = P/4

The width of the enclosure that maximizes the area is 1/4 the length of the available fence.

Using 100 feet of fence, the dimensions are ...

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Additional comment

Note that we have solved this in a generic way. The solution given is the general solution to the 3-sided enclosure problem.

This is a special case of the general rectangular enclosure problem which has the solution that the total cost in one direction is equal to the total cost in the orthogonal direction. This rule applies even when costs are different for the different sides or for any partitions that might divide the enclosure.

Here the "cost" is simply the length of the fence. The 50 ft of fence parallel to the barn is equal in length to the 25 +25 ft of fence perpendicular to the barn.

The dimensions that give the maximum area inside the rectangle are x = 50 feet (parallel to the barn wall) and y = 25 feet (perpendicular to the barn wall).

To maximize the area of the rectangular fence, follow these steps:

1. Let's assign variables to the dimensions: let the length of the rectangle parallel to the barn wall be x feet, and the length perpendicular to the barn wall be y feet.

2. We are given that 100 feet of fence will be used for the other three sides. This means the fencing equation is:

x + 2y = 100.

3. Solve for x:  x = 100 - 2y.

4. The area A of the rectangle is given by the product of its dimensions: A = xy.

5. Substitute the expression for x from step 3 into the area formula:  A = (100 - 2y)y.

6. Expand the expression: A = 100y - 2y^2.

7. To maximize the area, we need to find the maximum value of the quadratic function A(y). Since the coefficient of the y^2 term is negative, the graph of A(y) is a downward-opening parabola, which means it has a maximum value.

8. To find the maximum, we'll use the vertex formula for parabolas: y_vertex = -b/(2a), where a = -2 and b = 100. Plugging in these values, we get y_vertex = -100/(2 * -2) = 25.

9. Substitute the value of y_vertex back into the equation for x: x = 100 - 2(25) = 50.

10. So the dimensions that give the maximum area inside the rectangle are x = 50 feet (parallel to the barn wall) and y = 25 feet (perpendicular to the barn wall).

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(a) The average number of potential customers entering the system each hour is 2.5 customers per hour.

(b) The probability that the server will be busy is equal to the utilization factor, which is 0.67.

a) The average number of potential customers entering the system each hour can be calculated using Little's law, which states that the long-term average number of customers in a stable system is equal to the long-term average arrival rate multiplied by the long-term average time a customer spends in the system.

In this case, the arrival rate is 3 customers per hour, and the average time a customer spends in the system is equal to the average time between arrivals plus the average service time, which is 1/3 + 1/2 = 5/6 hour.

Therefore, the average number of potential customers entering the system each hour is 3 x 5/6 = 2.5 customers per hour.

b) The probability that the server will be busy can be calculated using the formula for the utilization factor, which is equal to the long-term average service rate divided by the system capacity.

In this case, the service rate is 2 customers per hour, and the system capacity is 3 customers. Therefore, the utilization factor is 2/3 = 0.67. The probability that the server will be busy is equal to the utilization factor, which is 0.67.

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