Review question

# Can we maximise the volume of this metal tin? Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

Ref: R6180

## Solution

A cylindrical tin and its tight-fitting cylindrical lid (overlapping the tin by $\quantity{2}{cm}$) are to be made from thin sheet metal. When made, the total area of sheet metal used is $\quantity{350\pi}{cm^2}$. The base radius of the tin is $\quantity{x}{cm}$ and its height is $\quantity{y}{cm}$ as shown in the diagram. Show that, neglecting the thickness of the metal, $x^2+xy+2x=175.$

We know that the surface area of the tin is $\quantity{350\pi}{cm^2}$. In terms of $x$ and $y$, the surface area of the tin is: \begin{align*} 350\pi &=\text{area of base} +\text{area of side}+\text{area of side of lid}+\text{area of top of lid} \\ &= \pi x^2+2\pi x y+2\pi x \times2+\pi x^2 \\ &= 2\pi x^2+2\pi xy +4\pi x, \end{align*} and dividing through by $2\pi$ we get $$$\label{eq:1} 175=x^2+xy+2x.$$$

Deduce that the volume, $\quantity{V}{cm^3}$, of the tin is given by $V= \pi(175x-2x^2-x^3).$

The volume of the tin is $V= \pi x^2 y.$

We can eliminate $y$ by rearranging equation $\eqref{eq:1}$ to find that $xy=175-x^2-2x$, and so $V=\pi x^2 y=\pi x(xy)=\pi x(175-x^2-2x)=\pi(175x-x^3-2x^2).$

If $x$ may vary, find the values of $x$ and $y$ for which $V$ has its maximum value.

We differentiate $V$ with respect to $x$, giving $\frac{dV}{dx}=\pi(175-3x^2-4x)=0,$ so to find stationary points of $V$, we must solve $3x^2+4x-175=0.$

We can factorise this equation as $(3x+25)(x-7)=0$ (or use the quadratic formula).

Since $x$ is the radius of the tin, it must be positive, so $x=7$.

We have that $\frac{d^2V}{dx^2}=\pi(-6x-4),$ which is negative when $x$ is positive, so $x=7$ gives a local maximum for $V$.

Furthermore, as there are no other stationary points for $V$ with $x>0$, this must give the maximum possible value for $V$ for $x\ge0$.

We need to find the corresponding $y.$ We saw in the previous part that $xy=175-x^2-2x,$ so that $7y=175-49-14=112,$ and hence $y=16$.