l+w+h=96 Packing Algebra Problem

Question

l+w+h=96

A) What is the formula for the volume of the box, in terms of l, w, and h? what’s the formula for volume of a box?

Let’s simplify this case a little more. Suppose we have a box with a square base, so let x= length = width. 

B) Solve for h in terms of x given the first equation, and then determine the function for the volume of the box ONLY in terms of x. 

C) Draw the graph of the function below, using an appropriate window. Make sure you have two x-intercepts and a maximum in you window. 

D) Using the graph, what dimensions (specific values of x and h) will yield the maximum volume?

Answer 1

The equation l + w + h = 96 is a constraint on the dimensions of your box.

A) From 4th grade math, the volume (V) of a box is V = l·w·h

B) Let l = w = x, so the base of the box is a square. The first equation becomes x + x + h = 96, or 2x + h = 96. Solving for h, we get h = 96 - 2x. So for volume we get:

V = l·w·h

V = x·x·(96-2x)

V = x²(96 - 2x) = -2x²(x - 48)

C) Graph it on your graphing calculator or a graphing app. The y-axis will be the volume V. The intercepts are at x = 0 and x = 48, the maximum occurs at x = 32. The maximum value on the y-axis is 32,768 so size your graphing window accordingly.

D) From the graph, the maximum occurs at (32, 32768). This means the maximum volume of the box is 32,768 units³ and occurs when x = l = w = 32 units. h = 96 - 2x = 96 - 64 = 32 units. The maximum volume will occur when the box is a cube with all sides being 32 units long.