Question

problem 7
the figure is a diagram of a wall. lengths are given in feet.
a. how many square feet of wallpaper would be needed to cover the wall? explain your reasoning.
b. wallpaper is sold in rolls that are 2 feet wide. what is the minimum length you would need to purchase to cover the wall?

Explanation:

Step1: Divide the wall into shapes

The wall can be divided into a rectangle and a triangle. The rectangle has length 48 feet and height 9 feet. The triangle has base 48 feet and height 6 feet.

Step2: Calculate rectangle area

The area formula for a rectangle is $A = l\times h$. So, $A_{rectangle}=48\times9 = 432$ square - feet.

Step3: Calculate triangle area

The area formula for a triangle is $A=\frac{1}{2}\times b\times h$. So, $A_{triangle}=\frac{1}{2}\times48\times6=144$ square - feet.

Step4: Calculate total wall area

The total area of the wall $A = A_{rectangle}+A_{triangle}=432 + 144=576$ square - feet. This answers part a.

Step5: Calculate length of wallpaper roll

The area of the wallpaper roll is $A_{roll}=l\times w$, where $w = 2$ feet. We know the area of the wall is 576 square - feet. We want to find $l$ when $A_{roll}=576$ and $w = 2$. Using the formula $l=\frac{A}{w}$, we get $l=\frac{576}{2}=288$ feet.

Answer:

a. 576 square feet. The wall is composed of a rectangle with area $48\times9 = 432$ square feet and a triangle with area $\frac{1}{2}\times48\times6 = 144$ square feet. The total area is $432+144 = 576$ square feet.
b. 288 feet. Since the area of the wall is 576 square feet and the width of the wallpaper roll is 2 feet, using the formula $l=\frac{A}{w}$, the length of the roll needed is $\frac{576}{2}=288$ feet.