Minimise material in a thick lidless container · source reference

Printed question

A manufacturer wants to design a lidless container with a square base and a capacity of $1000\,\mathrm{cm}^3$. Given that the uniform thickness of the container is 1 cm, using differentiation, find the internal length of the tank, $x$ cm, such that the volume of the material used is minimised. [6]

[2009/MJC/Prelim/I/11]

Printed working

Let internal height be $h$ cm, external length $Y$ cm and external height $Z$ cm. Container capacity gives $x^2h=1000\Rightarrow h=1000/x^2$. Also $Y=2+x$, $Z=1+h$.

Material needed:

$$\begin{aligned}S&=Y^2Z-x^2h\\&=(2+x)^2\left(1+\frac{1000}{x^2}\right)-x^2\left(\frac{1000}{x^2}\right)\\&=x^2+4\left(\frac{1000}{x^2}+1\right)(x+1).\end{aligned}$$

At maximum/minimum, $dS/dx=0$:

$$2x+4(x+1)\left(-\frac{2000}{x^3}\right)+4\left(\frac{1000}{x^2}+1\right)=0,$$ $$2x^4+4x^3-4000x-8000=0.$$

$x=-2$ is rejected since $x>0$, or $x=12.6$ (3 s.f.).

$$\frac{dS}{dx}=2x-\frac{8000}{x^2}-\frac{8000}{x^3}+\frac{4000}{x^2}+4=2x+4-\frac{4000}{x^2}-\frac{8000}{x^3},$$ $$\frac{d^2S}{dx^2}=2+\frac{8000}{x^3}+\frac{24000}{x^4}.$$

At $x=12.6$, $S''=2+8000/12.6^3+24000/12.6^4>0$. Thus material volume is minimised at $x=12.6$ cm (3 s.f.). The quartic factors $2(x+2)(x^3-2000)$, giving exact $x=\sqrt[3]{2000}$.

Teacher notes

Express $S$ in terms of only one variable, $x$.

Need to verify using a first or second derivative test that the material volume is minimised at $x=12.6$, (1) even though it is the only value found, (2) unless the question states there is no need to do so.

Answer key

$x=12.6$ (3 s.f.).

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