Question

jill has 120 feet of fencing to use for two horse paddocks. to increase the area she can enclose, she plans on using the side of the barn as one side of her paddock as shown below.
image of a rectangle labeled paddock with the top side labeled barn
determine which equation can be used to find the dimensions of the horse paddock when the area is 1,800 square feet. let x represent the width of the paddock.
a. $2x^2 - 120x = 1,800$
b. $x^2 + 120x = 1,800$
c. $x^2 - 120x = 1,800$
d. $-2x^2 + 120x = 1,800$

Explanation:

Step1: Analyze the paddock's dimensions

The paddock uses the barn as one side, so the fencing is for two widths and one length. Let \( x \) be the width. Then the length \( L = 120 - 2x \) (since total fencing is 120 feet, used for 2 widths: \( 2x \), so length is remaining fencing).

Step2: Recall the area formula for a rectangle

Area \( A = \text{length} \times \text{width} \). Given \( A = 1800 \), width \( = x \), length \( = 120 - 2x \). So \( A = x(120 - 2x) \).

Step3: Expand the area formula

\( x(120 - 2x) = 1800 \) expands to \( 120x - 2x^2 = 1800 \), which rearranges to \( -2x^2 + 120x = 1800 \).

Answer:

D. \( -2x^2 + 120x = 1800 \)