Question

A rectangular box has a base that is 4 times as long as it is wide. The sum of the height and the girth of the box is 200 feet. (a) Express the volume V of the box as a function of its width w. Determine the domain of V (w).

Answer 1

(a)
`V = (-8W^3 + 800W^2)/3`

(b)
`W > 100`

Explanation:

Let's call the length of the box L, the width W and the height H. Then, we can write the following equations:

"A rectangular box has a base that is 4 times as long as it is wide"

`L = 4W`

"The sum of the height and the girth of the box is 200 feet"

`H + (2W + 2H) = 200`

`2W + 3H = 200 \rightarrow H = (200 - 2W)/3`

The volume of the box is given by:

`V = L * W * H`

Using the L and H values from the equations above, we have:

`V = 4W * W * (200 - 2W)/3`

`V = (-8W^3 + 800W^2)/3`

The domain of V(W) is all positive values of W that gives a positive value for the volume (because a negative value for the volume or for the width doesn't make sense).

So to find where V(W) > 0, let's find first when V(W) = 0:

`(-8W^3 + 800W^2)/3 = 0`

`-8W^3 +800W^2 = 0`

`W^3 -100W^2 = 0`

`W^2(W -100) = 0`

The volume is zero when W = 0 or W = 100.

For positive values of W ≤ 100, the term W^2 is positive, but the term (W - 100) is negative, then we would have a negative volume.

For positive values of W > 100, both terms W^2 and (W - 100) would be positive, giving a positive volume.

So the domain of V(W) is W > 100.