Question

4. Rachel has 3000 feet of fencing to enclose a rectangular field for her horses. One side of
the field lies along the river, so only three sides require fence material.
a. Express the area A of the rectangle as a function of the width x of the rectangle.
b. What is the domain of A?
c. Use a graphing calculator to find the value of x that produces the largest area.
I

Answer 1

A is the correct answer

Answer 2

The area of the field is the amount of space on the field.

• The area as a function of x is $A(x) = 1500x - 0.5x^2$.
• The domain is x > 0
• The value of x that produces the largest area is 1500

The perimeter of the field is given as:

$P = 3000$

Because one side of the field will not be fenced, the perimeter is calculated as:

$P = w + 2l$

Where

$w = x$

So, we have:

$3000 = x + 2l$

Solve for 2l

$2l = 3000 - x$

Solve for l

$l = \frac{3000 -x}{2}$

The area of the field is

$A = l \times w$

So, we have:

$A =\frac{3000 -x}{2} \times x$

$A =(1500 - 0.5x) \times x$

$A = 1500x - 0.5x^2$

Represent as a function

$A(x) = 1500x - 0.5x^2$

(b) The domain

The width of the field must be greater than 0.

So, the domain of A is:

$x > 0$

(c) The largest Area

$A(x) = 1500x - 0.5x^2$

Differentiate

$A'(x) = 1500 - x$

Set to 0

$1500 - x = 0$

Solve for x

$x = 1500$

Hence, the value of x that produces the largest area is 1500

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