[填空题]
Engineers at a laborhcr +tm1 ;dw*bj9f w/z 6jqws6 ;pu 7dl6b6lajfd/* xiatory are designing a new type of gas storage container. The design consists of a cone with radius and vertical height r, on top o/i7 d ;fu6xlapbl dj6*f 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$$r^2$;a= ,b= .
2. Find an expression for the surface area of the container, A , in terms of r, $\ell$ and $\pi$ .
A=$({\sqrt{2}}+a)$$\pi$$r^2$+b$\pi$rℓ;a= ,b= .
3. Given the design constraint $\ell=\frac{10-2 \pi r^{2}}{\pi r} $, show that V=$10 r-\frac{5 \pi r^{3}}{3}$ .
V=ar-$\frac{b\pi r^3}{3}$;a= ,b= .
4. Find $ \frac{\mathrm{d} V}{\mathrm{~d} r}$ .
$\frac{dV}{dr}$=a-b$\pi$$r^2$;a= ,b= .
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= .
6. Find the length of the cylinder, $ \ell $, for which V is a maximum.
7. Calculate the maximum volume, V , of the container.
