Hello, Kali!

A rectangular swimming pool is to be built with an area of 1800 ft².

The owner wants 5-ft wide decks along either side and 10-ft wide decks at the two ends.

Find the dimensions of the smallest piece of property on which this pool can be built. Code:

: 10 : - - - L - - - : 10 :
- * - - * - - - - - - - * - - * -
5 | | :
- * * - - - - - - - * * :
: | | | | :
: | | | | :
W | | W | | W+10
: | | | | :
: | | L | | :
- * * - - - - - - - * * :
5 | | :
- * - - * - - - - - - - * - - * -
: - - - - - L+20 - - - - - :

The dimensions of the pool are $\displaystyle L$ by $\displaystyle W.$

The area of the pool is 1800 ft²: .$\displaystyle LW \:=\:1800 \quad\Rightarrow\quad W \:=\:\tfrac{1800}{L}$ .[1]

The total area has dimensions $\displaystyle L+20$ by $\displaystyle W+10$

We have: .$\displaystyle A \;=\;(L+20)(W+10) \;=\;LW + 10L + 20W + 200$ .[2]

Substitute [1] into [2]: .$\displaystyle A \;=\;L\left(\tfrac{1800}{L}\right) + 10L + 20\left(\tfrac{1800}{L}\right) + 200 $

. . which simplifies to: .$\displaystyle A \;=\;10L + 36000L^{-1} + 200$

Differentiate and equate to zero: .$\displaystyle 10 - 36000L^{-2} \:=\:0$

Multiply by $\displaystyle L^2\!:\;\;10L^2 - 36000 \:=\:0 \quad\Rightarrow\quad 10L^2 \:=\:36000 \quad\Rightarrow\quad L^2\:=\:3600$

. . Hence: .$\displaystyle L \:=\:60$

Substitute into [1]: .$\displaystyle W \:=\:\frac{1800}{60} \quad\Rightarrow\quad W\:=\:30$

Therefore, the property will be: .$\displaystyle \begin{Bmatrix}L+20 &=& 80\text{ ft} \\ & & \text{by} \\W+10 &=& 40\text{ ft} \end{Bmatrix} $