Chapter 4.1, Problem 54E

To determine

To find:width of the sidewalk that surrounds the pool.

Answer to Problem 54E

Width of the sidewalk = 5 feet

Explanation of Solution

Given information: Construction: The Santa Fe Recreation Department has a 50-foot by 70-ffot area for construction of a new public swimming pool. The pool will be surrounded by a concrete sidewalk of constant width. Because of water restrictions, the pool can have a maximum area of 2400 square feet.

  

Calculation:

Dimensions of the total area $=70\text{ft}\ast 50\text{ft}$

Let width of a concrete sidewalk be x ft.

The maximum area the pool can have, 2400 square feet.

Since it is surrounded by an x ft width concrete sidewalk

Dimensions of the pool $70-2x\text{ and }50-2x$

The maximum area the pool can have, 2400 square feet.

  $\left(70-2x\right)\left(50-2x\right)=2400$

  $\begin{array}{l}3500-100x-140x+x^2=2400\\ 3500-240x+4x^2=2400\\ 4x^2-240x+1100=0\\ x^2-60x+275=0\\ \left(x-55\right)\left(x-5\right)=0\\ x=55\text{ or }x=5\end{array}$

Since the width of the side walk cannot be 55 feet on both sides, x cannot take a value of 55. So x = 5.

Width of the sidewalk = 5 feet