For a math homework assignment, Karla found the area and perimeter of a room of her house. She reported that the area of her rectangular living room is 180 square feet and that the perimeter is 54 feet. When drawing a sketch of her living room the next day, she realized that she had forgotten to write down the dimensions of the room. What are the dimensions of Karla’s living room, in feet?
The correct answer is C.
You can solve this problem by solving the system of equations 2w + 2l = 54 and wl = 180, which result from using the perimeter and the area formulas, respectively.
2w + 2l = 54
w + l = 27
Solving wl = 180, l = 180/w
Substituting this value for l, you obtain
w + 180/w = 27
Multiplying both sides by w,
w2 + 180 = 27w
w2 - 27w + 180 = 0
Factoring the quadratic equation,
(w - 12)(w - 15) = 0
w = 12 or w = 15
The corresponding solutions for l are 180/w = 15 and 180/w = 12.
The system of equations is symmetric in w and l, so that the solutions for either w or l give the dimensions of the living room.