Let

w = the width, w>=0

l = the length, l>=0

d = 15 in the diagonal

A = 108 in^2 the area

Since A = length*width we have:

l*w = 108

l = 108/w

Using the Pythagorean Theorem we have:

d^2 = l^2 + w^2

Plug d = 15 and l = 108/w into the last equation:

15^2 = (108/w)^2 + w^2

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click here to see the equation solved for the width w

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w1 = 9 in

w2 = 12 in

For w = 9 in we have l = 108/9 that is l = 12 in.

For w = 12 in we have l = 108/9 that is l = 9 in.

Since we assume that the length is longer that the width we conclude:

The length of the rectangle is 12 inches while the width is 9 inches.