Question

A rectangular prism has a base area of 56 square feet and a volume of 840 cubic feet. The length of the base is longer than the width. If the sum of the length and width of the base is equal to the height of the pyramid, what is the length of the base in feet?

Answer 1

The answer is 8 ft.

The base area of rectangular prism is: A = l * w = 56 ft²
The length of the base is longer than the width: l > w

The volume of the prism is: V = l * w * h = 840 ft³
The sum of the length and width of the base is equal to the height of the pyramid: l + w = h

So:
l * w = 56
l * w * h = 840
___
56 * h = 840
h = 840 / 56
h = 15 ft

Now, we know that
l + w = 15         ⇒ w = 15 - l
l * w = 56
___
l * (15 - l) = 56
15l - l² = 56
0 = l² - 15l + 56

Or: l² - 15l + 56 = 0

Let's solve the quadratic function:
l = (-b +/-√(b² - 4ac)/(2a)
  = (15 +/-√(-15)² - 4 * 1 * 56))/(2*1)
  = (15 +/- √(225 - 224))/2
  = (15 +/- √1)/2
  = (15 +/-1)/2
 
l = (15+1)/2 = 16/2 = 8
or
l = (15-1)/2 = 14/2 = 7

If l = 8, then w = 15 - 8 = 7.    So, l > w
If l = 7, then w = 15 - 7 = 8.     So, l < w

Therefore, l = 8 ft.