Chapter 8.3, Problem 44PPS

To determine

To find: Whether there is enough room for Eva and her friends.

Answer to Problem 44PPS

Yes! There is enough room.

Explanation of Solution

Given information:

Formula used: $\theta =\frac{\left(n-2\right)180}{n}$

Calculation:

Measuring an interior angle of a regular polygon,

  $\begin{array}{l}\theta =\frac{\left(6-2\right)180}{6}\\ \theta =\frac{120°}{2}\\ \theta =60°\end{array}$

Therefore, the triangles will be $30-60-90$ special right triangle with hypotenuse of 7.

  $\begin{array}{c}2S=7\\ S=3.5\\ 3.5\sqrt{3}=7\end{array}$

The rectangle has side length of $7$ and a.

  $\begin{array}{c}a=2L\\ a=2\cdot 3.5\sqrt{3}\\ a=7\sqrt{3}\end{array}$

The area of one triangle,

  $\begin{array}{l}A=\frac{1}{2}\times 3.5\times 3.5\sqrt{3}\\ A\approx 10.61\end{array}$

Area of the rectangle,

  $\begin{array}{l}A=7\times 7\sqrt{3}\\ A\simeq 84.87\end{array}$

Total area of four triangles and one rectangle,

  $4\times 10.61+84.87=127.31f{t}^2$

Since, each guest take $8sq.ft$ ,

  $\therefore \frac{127.31}{8}=15.9\text{guests}$

  $\therefore 15>12\text{friends}+\text{Eva}$

Therefore, there is enough room.