The area of a rectangle is 72 square inches and the perimeter is 44 inches. Find its dimensions.

Question:

The area of a rectangle is {eq}72 {/eq} square inches and the perimeter is {eq}44 {/eq} inches. Find its dimensions.

The Perimeter and Area of a Rectangle:

A rectangle is a quadrilateral with two pairs of parallel sides. The parallel sides in a rectangle are equal and each of the interior angles are right angles. We calculate the perimeter of a rectangle by adding the lengths of all the sides. The area of a rectangle, on the other hand, is calculated by multiplying the length by the width.

Answer and Explanation:


The area of a rectangle is given by:

  • {eq}A = l/\times w {/eq}

And the perimeter is given by:

  • {eq}P = 2(l + w) {/eq}

If the area is 72 square inches and the perimeter is 44 inches, then:

  • {eq}72 = l\times w {/eq}......................................................................................(i)
  • {eq}44 = 2(l + w) {/eq}
  • {eq}22 = l + w {/eq}......................................................................................(ii)

Solving for l in equation (ii) and substituting it into equation (i):

  • {eq}l = 22 - w {/eq}
  • {eq}72 = w(22 - w) {/eq}
  • {eq}72 = 22w - w^2 {/eq}
  • {eq}w^2 - 22w + 72 = 0 {/eq}

Solving the quadratic equation by factorization, we have:

  • {eq}w^2 - 18w - 4w + 72 = 0 {/eq}
  • {eq}w(w - 18) - 4(w - 18) = 0 {/eq}
  • {eq}(w - 4)(w - 18) = 0 {/eq}
  • {eq}w = 4\, in, \quad w = 18\, in {/eq}

The rectangle will have two possible dimensions.

When {eq}w = 4\, in {/eq}, the length is equal to {eq}l = 22 - 4 = 18\, in {/eq}.

When {eq}w = 18\, in {/eq}, the length is equal to {eq}l = 22 - 18 = 4\, in {/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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