2005 PMWC Problems/Problem I13

Problem

Sixty meters of rope is used to make three sides of a rectangular camping area with a long wall used as the other side. The length of each side of the rectangle is a natural number. What is the largest area that can be enclosed by the rope and the wall?

Solution

Let $l$ be the side parallel to the wall, and $w$ adjacent to the wall. Then $l + 2w = 60 \Longrightarrow l = 60 - 2w$. The area of the rectangle is $lw = -2w^2 + 60w$; this is a quadratic equation, which we can maximize using the formula $\frac{-b}{2a} = 15$. Hence the area is $-2(15)^2 + 60(15) = 450$.

See also

2005 PMWC (Problems)
Preceded by
Problem I12
Followed by
Problem I14
I: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
T: 1 2 3 4 5 6 7 8 9 10
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