High School

If a farmer has 18,000 linear feet of fencing with which to construct his pens, what is the largest possible area he can enclose?

Set up the equation:

A. \(18000 = x + y\)
B. \(18000 = 4x + 27\)
C. \(18000 = 2x + 4y\)
D. \(18000 = 3x + 2y\)
E. \(18000 = 2x + 3y\)

Calculate the maximum area:

A. \(A(x) = 4500x - 27\)
B. \(A(x) = 4500x - \frac{1}{3} x^2\)
C. \(A(x) = 60000 - 2x^2\)
D. \(A(x) = 6000x - 2x^2\)
E. \(A(x) = 9000x - 2\)

Answer :

1. The largest possible area he can enclose is 18000 = x + y. So, the correct answer is A. 18000 = x + y

2.the maximum area is A(x) = 4500x² . So, the correct answer is A.

A(x) = 4500x²

Let's break down the problem step by step.

Setting up the equation:The farmer has 18,000 linear feet of fencing to construct a pen. Let's assume the length of the pen is x feet and the width is y feet. The perimeter of the pen (which is the total length of fencing used) can be represented by the equation: Perimeter = 2x + 2y

Since the total length of fencing is 18,000 feet, we can write the equation as: 2x + 2y = 18000

Now, we can simplify this equation by dividing both sides by 2: x + y = 9000

So, the correct equation is 18000 = x + y, which corresponds to option A.

2.Calculating the maximum area:The area of a rectangular pen can be calculated using the formula: Area = Length × Width

Area = x × y

We want to maximize the area while keeping the perimeter (fencing length) constant at 18,000 feet. From the first equation, we know that x + y = 9000, so we can solve for y: y = 9000 - x

Substituting this value of y into the area formula: Area = x × (9000 - x)

Now we have an equation for the area in terms of x. To find the maximum area, we need to find the value of x that maximizes this equation. This is a quadratic equation, and the maximum occurs at the vertex of the parabola.

The formula for the x-coordinate of the vertex of a parabola of the form ax² + bx + c is given by: x_vertex = -b / (2a)

In our case, the equation for the area is -x² + 9000x. Comparing this to ax² + bx + c, we have a = -1, b = 9000, and c = 0. Plugging these values into the formula: x_vertex = -9000 / (2 * -1)

x_vertex = 4500

So, the maximum area occurs when x = 4500 feet.

Now, we can calculate the corresponding y value using the equation y = 9000 - x: y = 9000 - 4500

y = 4500

Therefore, the maximum area is:Area = x × y

Area = 4500 × 4500

Area = 20,250,000 square feet

So, the correct equation for the maximum area is A(x) = 4500x² which corresponds to option A.

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Correct Question is:

31 wall river 2 set up the equation : If a farmer has 18,000 linear it of fencing with which to construct his pens,

1. what is the largest possible area he can enclose ?

A. 18000 = x + y

B. 18000 = 4x +27

C. 18000=2x+4y

D. 18000 = 3x+2y

E. 18000=2x+3y 3

2. calculate the max area, area=

A. A(x) = 4500x-27

B. A(x) = 4 500x - 1 3 x 2

C.A(x) = 60000 - 2x²

D. A(x) = 6000x - 2 x ²

E. A(x) = 9000x-2 ta २ 3 2 2 .ft 2