Question

If $30,000c{m}^2$ of material is available to make a box with a square base and an open top, find the largest possible volume of the box.

Answer 1

Step 1: Calculation of the relation between height and base.

Suppose the length of one side of the base of the box is $x\mathrm{cm}$ and its height is $h\mathrm{cm}$.

The volume of the box is $h{x}^2$ and the surface area is $4hx+x^2$.

Equate $4hx+x^2$ with $30,000c{m}^2$ and simplify.

$$\begin{gathered} 4hx+x^2=30000 \\ \Rightarrow 4hx=30000-x^2 \\ \therefore h=\frac{7500}{x}-\frac{x}{4} \end{gathered}$$

Step 2: Calculation of volume in terms of base.

Substitute the expression of $h$ in the expression of volume and simplify.

$$\begin{gathered} v=h{x}^2 \\ \Rightarrow v=\left(\frac{7500}{x}-\frac{x}{4}\right)x^2 \\ \therefore v=7500x-\frac{1}{4}{x}^3 \end{gathered}$$

Step 3: Calculation of critical points.

Differentiate $v\left(x\right)$ with respect to $x$.

$v\text{'}\left(x\right)=7500-\frac{3}{4}{x}^2$

Solve $v\text{'}\left(x\right)=0$ for $x$.

$$\begin{gathered} v\text{'}\left(x\right)=0 \\ \Rightarrow 7500-\frac{3}{4}{x}^2=0 \\ \Rightarrow \frac{3}{4}{x}^2=7500 \\ \Rightarrow x^2=\frac{4}{3}\times 7500 \\ \Rightarrow x^2=10,000 \\ \therefore x=100 \end{gathered}$$

Step 4: Calculation of the height:

The box has the largest possible volume for $x=100\mathrm{cm}$.

Calculate the value of $h$ for $x=100$.

$$\begin{gathered} h=\frac{7500}{100}-\frac{100}{4} \\ \Rightarrow h=75-25 \\ \therefore h=50 \end{gathered}$$

Step 5: Calculation of volume.

Calculate the largest possible volume for $h=50,x=100$

$$\begin{gathered} v=h{x}^2 \\ \Rightarrow v=50{\left(100\right)}^2 \\ \therefore v=500,000 \end{gathered}$$

Hence, the largest possible volume is $500,000{\mathrm{cm}}^3$.