SOLUTION: What is the largest possible area of a rectangle with integer side length and a perimeter of 22?

Question 142358This question is from textbook McDougal littell Math
: What is the largest possible area of a rectangle with integer side length and a perimeter of 22? This question is from textbook McDougal littell Math

Found 2 solutions by josmiceli, solver91311:
Answer by josmiceli(19441) (Show Source): You can put this solution on YOUR website!
Let one side =
Let the other side=
Let area =
Let perimeter=

Now substitute into area formula

This is a parabola that has a peak. I know because
of the (-) sign in front of
The peak is exactly between the x-intercepts, or at
where and
because the equation is in the form

So, the max area is

Answer by solver91311(24713) (Show Source): You can put this solution on YOUR website!

Given P = 22, so

A little algebra gets us:

So

This equation graphs to a convex down parabola with vertex at , telling us that the maximum area is achieved when the length of the side is . The difficulty is that the problem asks for the maximum area of a rectangle with integer side lengths. The nearest integer smaller than is 5 and the nearest integer larger than is 6.

Substituting 5 for W in gives us , so obviously substituting 6 for W in will give us . Since 5 and 6 are the nearest integers to the side length that provides the maximum area (notice that all 4 sides would be making a square) these must be the required side lengths and the maximum area for integer side lengths is
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