Question

a rectangular pool is x feet long and y feet wide. the new design of the pool increases the length by 15% and the width by 10%. additionally, a 4 foot wide walkway is added around the pool.
write the simplest expression for the perimeter in feet, of the redesigned pool including the walkway

Explanation:

Step1: Find new length and width

The original length is \( x \) feet, increasing by 15% means the new length is \( x + 0.15x = 1.15x \) feet. The original width is \( y \) feet, increasing by 10% means the new width is \( y + 0.10y = 1.10y \) feet. Then, we add a 4 - foot wide pathway around the pool. So the total length including the pathway is \( 1.15x+4 + 4=1.15x + 8\) feet (4 feet on each side), and the total width including the pathway is \( 1.10y+4 + 4=1.10y + 8\) feet.

Step2: Calculate the perimeter

The formula for the perimeter \( P \) of a rectangle is \( P = 2\times(\text{length}+\text{width})\). Substituting the total length and total width into the formula, we get:
\[

$$\begin{align*} P&=2\times((1.15x + 8)+(1.10y + 8))\\ &=2\times(1.15x+1.10y + 16)\\ &=2\times1.15x+2\times1.10y+2\times16\\ &=2.3x + 2.2y+32 \end{align*}$$

\]

Answer:

The perimeter of the pool including the pathway is \( 2.3x + 2.2y + 32 \) feet.