Among all rectangles that have a perimeter of 188 , find the dimensions of the one whose area is largest. Write your answers as fractions reduced to lowest terms.

Answer 1

The two formulas of interest:

P = 2L + 2W

A = LW

Since P = 188,

188 = 2L + 2W

L + W = 94

Let's find a formula for A that depends only on one variable (L or W).

L = 94 - W

Substituting that into the formula for area:

A = (94 - W)W

A = -W² + 94W

Written in function notation:

A(W) = -W² + 94W

Recognize this as a quadratic function that has a graph as a downward opening parabola. That means its vertex is the high point (or the x-coordinate of the vertex generates the max area).

The x-coordinate of the vertex lies on the axis of symmetry which has an equation x = -b/2a.

So x = -94/-2 = 47.

In our case W = 47 generates the max area. If W = 47 and L = 94 - W, then L = 47 too. That means the rectangle generating the max area is actually a square, and its area will be 2209.

Note that you can also find the x-coordinate of the vertex by writing the function as:

A(W) = -W(W - 94)

The roots are W=0 and W=94. The axis of symmetry is the vertical line with an x-coordinate that is the average of the roots (47).