Expert: Bobby Soltani - 4/29/2006

Question
The number of cubic inches of volume in a rectangular box is 351. The number of square inches on the box's surface area is 327. The product of the number of inches in the width and height is one more than twice the number of inches in the length. Find the dimensions of hte box and give all possible solutions.
I would really appreciate it if you could help me with this problem by sunday if you could please.


Answer
Hi Anthony,

Let the length of the box be L, the width W, and the height H.

The surface area is equal to the sum of the areas of all six sides.  Since the sides that are opposite each other are the same, we have three unique dimensions.  They are:

LW for the top and bottom
LH for two sides
WH for the other two sides

So the Surface area is

SA = 2LW + 2LH + 2WH = 327

The volume is equal to

V = LWH = 351

The product of the number of inches in the width and height is one more than twice the number of inches in the length.

WH = 1 + 2L

Plug 1 + 2L in for WH in the equation for the volume

V = L(1 + 2L) = 351

351 = L + 2L^2
solve for L

2L^2 + L - 351 = 0

Using the quadratic equation, L = 13

Now plug 13 in for L in the first two equations.

SA = 26W + 26H + 2WH = 327

The volume is equal to

13WH = 351

WH = 27

Substitute 27 in for WH in the surface area equation.

327 = 26W + 26H + 54

273 = 26W + 26H

divide both sides by 26

10.5 = W + H
We also have from above that WH = 27, so we can write


W = 10.5 - H
substituting into the HW = 27 equation

H(10.5 - H) = 27

10.5H - H^2 - 27 = 0
Using the quadratic equation, we get H = 6, or 4.5

If H = 6, then W = 10.5 - H = 4.5
If H = 4.5, then W = 10.5 - H = 6

So the answers is
L = 13, H =6, W = 4.5  or
L = 13, H = 4.5, W = 6

This was a tricky problem with several steps.  Try and work through each step to understand how to solve it.  Let me know if you have any questions.  Good luck!

Bobby