Question

A movie theater wants to update their entryway. The renovation will impact a 90 degree segment of the building, including a portion of theaters nine and one, as well as the entryway. It is 300 feet from the center of the lobby to the outer walls of the building. Find the area of the segment that will be impacted by construction.

Answer 1

The sketch of the image is given below

The area of the segment can be obtained as follow

$\text{Area of segment=Area of sector-Area of triangle}$

The area of the sector can be obtained as follow:

$\begin{gathered} \text{Area of sector=}\frac{\theta}{360}\times\pi r^2 \\ \text{where} \\ \theta=90^0 \\ r=300\text{ f}eet \\ \pi=3.14 \end{gathered}$$\begin{gathered} \text{Area of sector=}\frac{90}{360}\times3.14\times300^2 \\ \\ \text{Area of sector=70650fe}et^2 \\ \end{gathered}$

The area of the triangle will be

$\begin{gathered} \text{Area of triangle=}\frac{1}{2}r^2\sin \theta \\ \Rightarrow\frac{1}{2}\times300^2\times\sin 90 \\ \Rightarrow45000\text{feet}^2 \end{gathered}$

Therefore the area of the segment will be

$\begin{gathered} \text{Area of segment= Area of sector-Area of triangle} \\ \text{Area of segment}=70650-45000=25650\text{feet}^2 \end{gathered}$