PSAT Math Multiple-Choice Question 759: Answer and Explanation

Question: 759

An interior designer is selling wood flooring to be used by his client for a new room. The client has already purchased a set length of trim, which goes between the edge of the wood flooring and the wall. The trim is straight and cannot be curved, yet it can be joined to make right angle corners. The client does not wish to purchase any more trim and would like to use all of the trim when building the new room. If the interior designer wants to maximize the amount of wood flooring that the client purchases while satisfying the client's requirements, what should be the relationship between the length (L) and width (W) of the room's dimensions?

• A. L = W
• B. L = 2W
• C. W = L²
• D. L = W³

Correct Answer: A

Explanation:

(A) In order to maximize the area of flooring while minimizing the floor's perimeter, a square floor would be the best choice. A square always has at least as much and typically more area for a particular perimeter than a rectangle of the same perimeter. Therefore, the length and width should be equivalent.

To see this, try using concrete numbers. If you have a square and a rectangle, each with a perimeter of 20 units, the length of each side for the square must be 5 and the lengths of the sides of the rectangle could be a wide range of possibilities, such as 2, 8, 2, 8. The area of the square with a side of 5 is 5² = 25. The area of the rectangle is 2 × 8 = 16, which is much less than the area of the square. You can try this with other sample values for the rectangle's sides. However, you will consistently find that having the sides equivalent will lead to the greatest possible area.