Expert: Paul Klarreich - 11/16/2009

Question
I have two rectangles, one inside of the other.  The first one has dimensions of x+9 and x+8.  The inside rectangle has dimensions of 2x+1 and x-3.  The question is what is the maximum shaded area??  I already figured out that x must be greater than 25 or less than -3, but I have no idea what the maximum area would be.  I plugged the equation into my calculator, but couldn't find the max; the line looked linear rather than quadratic.  What should I do??

Answer
Questioner: Katharine
Country: United States
Category: Advanced Math
Private: No
Subject: pre-calculus
Question: I have two rectangles, one inside of the other.  The first one

>> The outside one????

has dimensions of x+9 and x+8.  The inside rectangle has dimensions of 2x+1 and x-3.  The question is what is the maximum shaded area??  

>> what area gets shaded?

I already figured out that x must be greater than 25 or less than

-3, but I have no idea what the maximum area would be.  I plugged the equation into my

calculator, but couldn't find the max; the line looked linear rather than quadratic.  What

should I do??

.......................................
I shall assume you mean:

The outside one is  (x + 9)(x + 8)  and
the inside is (2x+1)(x-3)  and
the shaded area is the difference:

(x + 9)(x + 8) - (2x + 1)(x - 3)

= x^2 + 17x + 72 - [2x^2 - 5x - 3]

= x^2 + 17x + 72 - 2x^2 + 5x + 3

= - x^2 + 22x + 75

This would be a parabola opening downward, so its vertex is a maximum.

The vertex is at  x = 11   [Look up the formula for the vertex of a parabola.]

Substitute  x = 11 to get the actual max area.