Engineers at a laboratory are designing a new ty
x l gt7va:/(fik;wt1 ml+4i nqpe of gas storage container. The design c
xtm47iiwf: g+l(aq /v n ;lt1konsists of a cone with radius and vertical height r, on top of a cylinder with length ℓ, where r and ℓ are measured in meters. A diagram of the container is shown below.
1. Find an expression for the volume, V , of the container, in terms of r, $\ell and \pi$ .
V=$\frac{{\pi}r^a}{b}$+ℓ$\pi$$r^2$;a=
(数值) ,b=
(数值) .
2. Find an expression for the surface area of the container, A , in terms of r, $\ell$ and $\pi$ .
A=$({\sqrt{2}}+a)$$\pi$$r^2$+b$\pi$rℓ;a=
(数值) ,b=
(数值) .
3. Given the design constraint $\ell=\frac{10-2 \pi r^{2}}{\pi r} $, show that V=$10 r-\frac{5 \pi r^{3}}{3}$ .
V=ar-$\frac{b\pi r^3}{3}$;a=
(数值) ,b=
(数值) .
4. Find $ \frac{\mathrm{d} V}{\mathrm{~d} r}$ .
$\frac{dV}{dr}$=a-b$\pi$$r^2$;a=
(数值) ,b=
(数值) .
The engineers aim to maximise the volume of the container for the given design constraints.
5. Using your answer to part (d), show that V is a maximum when $r=\sqrt{\frac{2}{\pi}} $
r=$\sqrt{\frac{a}{\pi}} \mathrm{m}$;a=
(数值) .
6. Find the length of the cylinder, $ \ell $, for which V is a maximum.
7. Calculate the maximum volume, V , of the container.