Question

a rectangular park is x feet long and y feet wide. the new design of the park increases the length by 37% and the width by 12%. additionally, a 7 - foot wide pathway is added around the park. enter the simplest expression for the perimeter, in feet, of the redesigned park, including the pathway.

Explanation:

Step1: Find the new length

The original length is \( x \) feet. Increasing the length by 37% means the new length is \( x + 0.37x = 1.37x \) feet.

Step2: Find the new width

The original width is \( y \) feet. Increasing the width by 12% means the new width is \( y + 0.12y = 1.12y \) feet. Then, we add a 7 - foot wide pathway around the park. This adds \( 2\times7=14 \) feet to the total length (7 feet on each side) and \( 2\times7 = 14 \) feet to the total width (7 feet on each side). So the final length is \( 1.37x+14 \) and the final width is \( 1.12y + 14 \).

Step3: Calculate the perimeter

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

$$\begin{align*} P&=2\times((1.37x + 14)+(1.12y+14))\\ &=2\times(1.37x+1.12y + 28)\\ &=2.74x+2.24y + 56 \end{align*}$$

\]

Answer:

The perimeter of the redesigned park (including the pathway) is \( 2.74x + 2.24y+56 \) feet.