A rectangle has a length that is 5 inches greater than its width, and its area is 104 square inches. The equation (x + 5)x = 104 represents the situation, where x represents the width of the rectangle.

(x + 5)x = 104

x2 + 5x – 104 = 0

Determine the solutions of the equation. What solution makes sense for the situation?

x =

What are the dimensions of the rectangle?

width = inches

length = inches

For this case, we have a ractangle of area 104 square inches, they tell us that its length is 5 inches greater than the width. In addition, we have the following equation \((x + 5) x = 104\) where the variable "x" represents the width of the rectangle.

By definition, the area of a rectangle is given by:

\(A = l * x\)

Where:

l: It's the lenght

x: It is the width

\(A = 104\) square inches

For the width we have:

\((x + 5) x = 104\\x ^ 2 + 5x = 104\\x ^ 2 + 5x-104 = 0\)

We find the solutions of the equation by factoring, that is, we look for two numbers that when multiplied give as result -104 and when summed give as result +5. So, those numbers are +13 and -8.

\(13 * -8 = -104\\13-8 = + 5\)

So, we have:

\((x + 13) (x-8) = 0\)

The roots are:

\(x_ {1} = - 13\\x_ {2} = 8\)

The solution that makes sense for the width of the rectangle is: x_ {2} = 8

Thus, the width of the rectangle is x = 8 inches

If the thickness is 5 inches greater than the width, then:

\(l = 5 + 8\\l = 13\ inches.\)

Verifying the area, we have:

\(A = 13 * 8 = 104\) square inches

ANswer:

\(width = 8\ inches\\length = 13\ inches\)