A community center is building a haunted graveyard. The graveyard is rectangular, and they want it to be at least 1400 square ft in area. They have 180 ft of decorative fencing for the graveyard. If they use all of the fencing, what are the possible widths the graveyard could be?
A) Strictly between 20 and 70 ft
B) Less than 20 ft or more than 70 ft
C) Between 20 ft and 70 ft inclusive
D) Less than or equal to 20 ft or greater than or equal to 70 ft
E) None of these

Question

Quizplus · Accepted Answer

The perimeter of the rectangle is 2(length + width) = 180 ft. Let's denote the length as L and the width as W. So, 2(L + W) = 180, which simplifies to L + W = 90. The area of the rectangle is L*W ≥ 1400 square feet. To find the possible widths, we can express L in terms of W from the perimeter equation: L = 90 - W. Substituting this into the area inequality gives W(90 - W) ≥ 1400. This quadratic inequality can be solved to find the range of W. The solutions to the corresponding equation W(90 - W) = 1400 will give the critical points, which are the bounds of W for which the area is at least 1400 square feet. Solving this quadratic equation, we find that the product of W and (90 - W) is maximized when W is between 20 and 70, inclusive, because outside this range, the product (and thus the area) would be less than 1400 square feet. Therefore, the possible widths of the graveyard are between 20 ft and 70 ft inclusive.

A Community Center Is Building a Haunted Graveyard