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Suppose a farmer has 120 feet of fencing to make a rectangular barnyard enclosure. If the...

Question:

Suppose a farmer has 120 feet of fencing to make a rectangular barnyard enclosure. If the enclosure is x feet long, express the area A of the enclosure as a function of the length x.

The Area of a Rectangle:

A rectangle is a figure that has 4 sides. The opposite sides in a rectangle are equal and the adjacent sides are equal. The longer side of the rectangle is called the length and the shorter side is called the width. We calculate the area of a rectangle by multiplying the length by the width.

Answer and Explanation:


The area of a rectangle is given by:

  • {eq}A = lw {/eq}

And the perimeter is given by:

  • {eq}P = 2(l + w) {/eq}

Given that the perimeter is 120 feet, then we have:

  • {eq}120 = 2(l + w) {/eq}

If the length is {eq}x\, \rm ft {/eq} long, then we have:

  • {eq}120 = 2(x + w) {/eq}

Simplifying, we have:

  • {eq}60 = x + w {/eq}

Solving for the width {eq}w {/eq}:

  • {eq}w = 60 - x {/eq}

Therefore, the area of the rectangular barnyard is equal to:

  • {eq}A = lw = x(60 - x)\, \rm ft^2 {/eq}
  • {eq}A = \boxed{(60x - x^2)\, \rm ft^2} {/eq}

Learn more about this topic:

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Measuring the Area of a Rectangle: Formula & Examples

from Geometry: High School

Chapter 8 / Lesson 7
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