Chapter 12.1, Problem 47SPR

To determine

To find the area of fabric in square yards to cover both sides of the cushions.

Answer to Problem 47SPR

Total area of fabric $=3.567$ Sq. Yards

Explanation of Solution

Given Information:

There are seven congruent circular cushion with a diameter of 12 inches around a center cushion with a diameter of 20 inches as shown below:

  

An extra 3 inches of fabric should be addedto the fabric around each cushion.

Formula used:

Area of circle $=\pi r^2$

Diameter of centre cushion $=20$ inches

Radius of centre cushion $=\frac{20}{2}+3$

  $=13$ inches

∴ Area of centre cushion $=\pi {\left(13\right)}^2=169\pi$ sq. in

Diameter of each of seven circular cushions = 12 inches

Radius of each of seven circular cushions $=\frac{12}{2}+3=9$ inches

∴ Area of each of seven circular cushions $=\pi \left(9^2\right)=81\pi$ sq. in

There are seven congruent smaller cushions and one central cushion. Each cushion has 2 circular sides (which is the fabric on back side).

The total area of fabric = 7 (2 times the area of smaller cushion) + (2 times the area of central cushion)

  $A=7\left(2\times 81\pi \right)+\left(2\times 169\pi \right)$

  $\begin{array}{l}=7\left(162\pi \right)+338\pi \\ =1134\pi +338\pi \end{array}$

  $=1472\pi$ sq. in

To convert sq. in to square Yards divide by $\left(144\times 9\right)$

  $A=\frac{1472\pi }{144\times 9}$

  $=3.567$ Sq. Yards