Find dimensions of a box, given the surface areas of its sides

Question

The top of a rectangular box has an area of 108 square centimeters, the sides have areas of 96 square centimeters and 72 square centimeters. What are the dimensions of the box? (reposted question from Martha L)

Answer 1

Call the box's width X, its depth Y, and its height Z. These are the three unknowns. Thankfully we can also write three equations, just enough information to solve the problem!

Equation 1: XY = 108

Equation 2: YZ = 96

Equation 3: ZX = 72

Now multiply Equation 1 by Equation 2 and then divide by Equation 3.

(XY)*(YZ)/(ZX) = 108*96/72

This simplifies to Y^2 = 144, so Y = 12.

Similarly, you can multiply Equation 1 by Equation 3 and divide by Equation 2 to get

(XY)*(ZX)/(YZ) = 108*72/96

which simplifies to X^2 = 81, so X = 9.

Finally, multiply Equation 2 by Equation 3 and divide by Equation 1 to get

(YZ)*(ZX)/(XY) = 96*72/108

which simplifies to Z^2 = 64, so Z = 8.

In conclusion, the box's width, depth, and height are 12 cm, 9 cm, and 8 cm, respectively. Just to double check: 12*9 = 108, 12*8 = 96, and 9*8 = 72.

Hope this helps, please message me if you need any clarification or additional explanation!