How do I solve this problem in description?

Question

A rectangular banner advertising a homecoming dance is 8 feet long and 6 feet wide. The committee wants to increase visibility by making the banner larger. If each dimension is increased by the same number of feet, the area of the new rectangle formed is 32 square feet more than the area of the original rectangle. By how many feet was each dimension increased?

Answer 1

Since they are asking you to find the number of feet the dimensions were increased, and they were both increased by the same amount, it makes sense to assign a variable name to this number, and then see if the description in the problem leads to an equation (which I'm sure it will!).

 

So...

 

The original sign is 8 by 6, so its original area is 48.

 

The new dimensions are 8 + x and 6 + x, so the new area is (8 + x)(6 + x) = 48 + 14x + x², which we can rewrite as x² + 14 + 48.

 

And this new area is 32 square feet more than the original area of 48, so the new area is 80.

 

That gives us the equation:

 

x² + 14 + 48= 80

 

Let's solve it:

 

x² + 14 + 48= 80

x² + 14 - 32 = 0

(x + 16)(x - 2) = 0

 

x = -16 or x = 2

 

You have to rule out -16, because if you "increased" the dimensions by -16 (which is the same as "decreasing" them), the dimensions would become negative.  So that doesn't make sense.

 

So the dimensions were each increased by 2 feet -- resulting in a length of 10 and a width of 8, for an area of 80, as expected.