Question

If 1200 cm² of material is available to make a box with a square base and an open top, find the largest possible volume of the box.

Short answer

\(4000\;{{\mathop{\rm cm}\nolimits} ^3}\)

Step-by-step solution

Given Data

1) Area of the box is 1200 cm².

2) The box has a square base and an open top.

Determination of the Volume of the box

Let a be the side of the bottom and h be the height of the box.

Now, Volume of the box \(V = {a^2}h\)…..(1)

The surface area is, \(S = {a^2} + 4ah = 1200\)

Eliminate h from above equation as:

\(h = \frac{{1200 - {a^2}}}{{4a}}\)

Put \(h = \frac{{1200 - {a^2}}}{{4a}}\)in equation (1).

\(V = \frac{1}{4}\left( {1200a - {a^3}} \right)\)

To find maximum volume, differentiate V with respect to a and equate to 0.

\(\begin{aligned}{c}300 - \frac{3}{4}{a^2} &= 0\\\frac{3}{4}{a^2} &= 300\\{a^2} &= 400\\a &= 20\end{aligned}\)

This is a case of maximum since the second derivative is negative.

Put \(a = 20\) in \(h = \frac{{1200 - {a^2}}}{{4a}}\)to find h.

Therefore, \(a = 20\) and \(h = 10\) .

Find Maximum Volume, \(V = \frac{1}{4}\left( {1200a - {a^3}} \right)\) as:

\(\begin{aligned} V &= \frac{1}{4}\left( {1200 \times 20 - {{20}^3}} \right)\\ &= 4000\;{{\mathop{\rm cm}\nolimits} ^3}\end{aligned}\)

Thus, the required volume is \(4000\;{{\mathop{\rm cm}\nolimits} ^3}\).