Question

A company is promoting "dome homes" as the newest fad in housing. Unfortunately, a segment of the ceiling in the model home is leaking and must be replaced. If the diameter of the Home is 66 feet, determine the area of the segment between the kitchen and the family room that must be replaced.

Answer 1

Step 1

Write the formula for the area of a segment.

$\frac{\theta}{360}\times\pi r^2-\frac{1}{2}r^2\sin \theta$

Where

$\begin{gathered} \theta=120^o \\ r=\text{ radius = }\frac{diameter}{2}=\frac{66}{2}=33\text{feet} \\ \pi=3.14 \end{gathered}$

Step 2

Substitute and find the area of the required segment

$(\frac{120}{360}\times3.14\times33^2)-(\frac{1}{2}\times33^2\times\sin \text{ 120)}$$1139.82-471.55=668.27ft^2$

Hence the area of the segment between the kitchen and the family room that must be replaced is 668.27ft²