An artist intends to paint a 84-square-foot mural on a wall that is 18 feet long and 10 feet high. If the artist wants to leave a border of uniform width, how wide should the border be? ( Hint: Let $w$ represent the width of the border. Then $18 - 2 w$ represents the length of the mural.)
A) 1 foot
B) 6 foot
C) 2 foot
D) 14 foot
E) 13 foot

Question

An artist intends to paint a 84-square-foot mural on a wall that is 18 feet long and 10 feet high. If the artist wants to leave a border of uniform width, how wide should the border be? ( Hint: Let $w$ represent the width of the border. Then $18 - 2 w$ represents the length of the mural.) 
A) 1 foot 
B) 6 foot 
C) 2 foot 
D) 14 foot 
E) 13 foot

Quizplus · Accepted Answer

The area of the entire wall is $18\cdot10=180$ square feet. If we subtract the 84 square feet of the mural, we have 96 square feet remaining for the border.

Let $w$ be the width of the border in feet. Then the dimensions of the mural itself are $18-2w$ feet by $10-2w$ feet, and the area of the mural is $(18-2w)(10-2w) = 84$ square feet.

Expanding the product, we get $180-36w-20w+4w^2=84$, which simplifies to $4w^2-56w+96=0$. Factoring out 4, we get $4(w^2-14w+24)=0$, which factors further as $4(w-2)(w-12)=0$. The possible widths of the border are then $w=2$ or $w=12$.

Since $w$ represents the width of the border, it must be positive and less than $\frac{18}{2}=9$ feet (to avoid the border sticking out beyond the edges of the wall). Therefore, our only solution is $w=2$ feet.

An Artist Intends to Paint a 84-Square-Foot Mural on a Wall