Write a polynomial expression to model the square feet of concrete needed

Question

Problem: Installing a new concrete sidewalk around a swimming pool. The pool is 25 feet long by 15 feet wide and I need the sidewalk to be the same all the way around.

 

1. Write polynomial expression to model the square feet of concrete needed

 

2. Determine how much concrete is needed if the sidewalk is 

a. 4 feet wide

b. 5 feet wide

c. 6 feet wide

Answer 1

First, the area of the pool itself is 25*15 = 375 square feet. Once we build the sidewalk around it with width x, the area of the total enclosure (sidewalk + pool) will be (25+2x)*(15+2x)=4x²+80x+375. I got this expression by adding the width to each dimension twice, since the sidewalk has to be the same width all the way around. The area of the sidewalk itself is then this area minus the area of the pool:

 

4x²+80x+375-375=4x²+80x

 

To get the areas for parts a b and c, you just need to plug in 4,5 and 6 for x in this expression:

 

a. 4(4)²+80(4)=4(16)+320=64+320=384 square feet

 

b. 4(5)²+80(5)=4(25)+400=100+400=500 square feet

 

c. 4(6)²+80(6)=4(36)+480=144+480=624 square feet