Question

A cone-shaped paper drinking cup is to be made to hold 27 cm³ of water. Find the height and radius of the cup that will use the smallest amount of paper.

Short answer

The radius of the cup is \(r \approx 2.632\,\,cm\) and height of the cup is \(h \approx 3.722\,\,cm\).

Step-by-step solution

To find the surface area

Let\(r\)be the radius of the cone and\(h\)be the height of the cone.

Using Pythagorean theorem, we have,

Volume of cone is given by,

\(\begin{aligned}{c}V &= \frac{1}{3}\pi {r^2}h\,\\27 &= \frac{1}{3}\pi {r^2}h\,\\{r^2} &= \frac{{81}}{{\pi h}}\\r &= \frac{9}{{\sqrt {\pi h} }}\,\, \cdots \cdots \left( 1 \right)\end{aligned}\)

Surface area is given by,

\(S = \pi r\sqrt {{h^2} + {r^2}} \)

Using\(\left( 1 \right)\)in the above equation, we get,

\(\begin{aligned}{c}S &= \pi \frac{9}{{\sqrt {\pi h} }}\sqrt {{h^2} + {{\left( {\frac{9}{{\sqrt {\pi h} }}} \right)}^2}} \\S &= 9\sqrt {\frac{\pi }{h}} \sqrt {{h^2} + \frac{{81}}{{\pi h}}} \\S &= 9\sqrt \pi \sqrt {h + \frac{{81}}{{\pi {h^2}}}} \end{aligned}\)

To find the height and the radius of the cup

To simplify the things, using the quantity without square root and constant on the side. Minimizing this would minimize\(S\).

\({S_2} = h + \frac{{81}}{{\pi {h^2}}}\).

Differentiating this, we get,

\(\begin{aligned}{c}{{S'}_2} &= 1 + \left( { - 2} \right)\frac{{81}}{\pi }{h^{ - 3}}\\{{S'}_2} &= 1 - \frac{{162}}{{\pi {h^3}}}\end{aligned}\)

Solving for\({S'_2} = 0\), we get,

\(\begin{aligned}{l}1 - \frac{{162}}{{\pi {h^3}}} &= 0\\ \Rightarrow \pi {h^3} &= 162\\ \Rightarrow {h^3} &= \frac{{162}}{\pi }\\ \Rightarrow h &= \sqrt(3){{\frac{{162}}{\pi }}}\\ \Rightarrow h \approx 3.722\,\,cm\end{aligned}\)

Now, the second derivative is,

\(\begin{aligned}{l}S'' &= - \frac{{\left( { - 3} \right)\left( {162} \right)}}{{{h^4}}}\\ \Rightarrow S'' &= \frac{{486}}{{{h^4}}}\end{aligned}\)

which is positive.

Thus,\(h\)is minimum.

Let us calculate the radius.

\(\begin{aligned}{c}r &= \frac{9}{{\sqrt {\pi h} }}\\r &= \frac{9}{{\sqrt {\pi \left( {3.722} \right)} }}\\r \approx 2.632\,\,cm\end{aligned}\)

The cup is very small.

Hence, the radius of the cup is \(r \approx 2.632\,\,cm\) and height of the cup is \(h \approx 3.722\,\,cm\).