Lilia wants to have a circular pool put in her backyard. She wants the rest of the yard to be paved with concrete.

1. If her yard is a $50$ ft. by $30$ ft. rectangle, what is the radius of the largest pool that will fit in her yard?

   Step 1 (a):

   Draw the rectangle.

   Step 2 (a):

   Fit the largest possible circle into the rectangle (it does not have to be in the center).

   Answer (a):

   The radius of the pool is $15$ ft.

   

2. If the concrete is to be $8$ inches thick, and costs $\$2.39$ per cubic foot, what is the cost of putting in the concrete? No concrete will be used in the pool. (Reminder: Volume $=$ (Base Area $·$ Depth ).

   Step 1 (b):

   Find the area of the yard (rectangle) and the area of the pool (circle).

   Step 2 (b):

   Subtract the area of the pool from the area of the yard to find the area that will be paved.

   $\text{A}=1500−225\pi\approx793.14\text{ ft.}^2$

   Step 3 (b):

   $\text{Since 8 inches is } \frac{2}{3}\ \text{of a foot, multiply the area of the region to be paved by two thirds.}$

   $V = 793.14 \cdot \frac{2}{3} \approx 528.76 \text{ ft.}^{3}$

   Step 4 (b):

   Now multiply by the price per cubic foot to find the total cost.

   $C=528.76 \text{ ft.}^{3} \cdot 2.39 \frac{\text{dollars}}{\text{ft.}^{3}}$

   Answer (b):

   The total cost is $\$1,263.74$.