I have 104 feet to build a dog pen. What is the largest width to maximise the area? What is the...

Question:

I have 104 feet to build a dog pen. What is the largest width to maximize the area? What is the area?

{eq}A=52x-x^2 {/eq}

{eq}x= {/eq}width

Perimeter and Area:

The perimeter is the distance around a geometric figure while the area is the amount of space that is covered by the figure. For a rectangular figure, its perimeter is equal to the sum of the four sides of the figure, and the area is the product of the length and the width.

Answer and Explanation:


The perimeter of the rectangular pen will be calculated as:

  • {eq}P = 2(l + w) {/eq}

Given that we have 104 feet to build the dog pen, then:

  • {eq}104 = 2(l + w) {/eq}

Solving for l, we have:

  • {eq}52 = l + w {/eq}
  • {eq}w = 52 - w {/eq}

The area of the pen will be calculated as:

  • {eq}A = l\times w {/eq}

But {eq}l = 52 - w {/eq}. Therefore:

  • {eq}A = w(52 - w) = 52w - w^2 {/eq}

Maximizing the area:

  • {eq}\dfrac{dA}{dw} = 0 {/eq}
  • {eq}\dfrac{dA}{dw} = 52 - 2w = 0 {/eq}

Solving for w:

  • {eq}2w = 52 {/eq}
  • {eq}w = \dfrac{52}{2} = 26\, ft {/eq}

Therefore, the area of the dog pen is equal to:

  • {eq}A = 52(26) - 26^2 = \boxed{676\, 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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