[填空题]
A manufacturer makes chemicazur (fc-59sicet qsbxpe oy6* g) (t,*rnzo(;l transport vessels in the form of a cylinder w7xin*rtlx.dq+o 0og8 ith a hemispherical front. The vessel has a total length of 4 m. The base radius of both the cylinder and the hemispherical+ r l xg0not8*oix.7qd front is 1m.
1. Write down the length of the cylinder.
$l_c$= m.
2. Find the total volume of the vessel.
$V_t$≈ m$^3$.
A truck company is looking to optimize and improve the dimensions of the vessel. The new vessel will also be in the form of a cylinder with a hemispherical front. It will have a length of l $\mathrm{~m}$ and a base radius of r $\mathrm{~m}$ .
There is a design constraint such that l+4 r=8 $\mathrm{~m}$ . The manufacturer wants to maximize the volume of the vessel.
3. Write down the length of the new vessel, l , in terms of r ; l=a-br , a= ,b= .
4. Show that the volume, V $\mathrm{~m}^{3}$ , of the new vessel is given by
V=8 $\pi r^{2}-\frac{13 \pi r^{3}}{3}$
5. Using your graphic display calculator, find the value of r which maximizes the value of V . The manufacturer claims that the new vessel has a capacity 20 $\%$ greater than the capacity of the original vessel.
6. State whether the manufacturer's claim is correct. Justify your answer.
