[填空题]
A greenhouse is made a8qv)jneof ib*8;ht4xs5n,k b269j2re vwdl wgjavj73 z)* b in the shape of a half cylinder. It is constructed from a galvanized steel frame with a multiw)bjgvj3 w*7z aall polycarbonate sheeting. The steel frame consists of a rectangular base, four semicircular arches and three further support rods, as shown in the following diagram.
The semicircular arches have radius r and the support rods each have length l .
Let S be the total length of steel used in the frame of the greenhouse.
1. Write down an expression for S in terms of r, l and $\pi$ .
S= l+ r+ $\pi$r
The volume of the greenhouse is 37.5 $\mathrm{~m}^{3}$ .
2. Write down an equation for the volume of the greenhouse in terms of r, l and $\pi$ .
3. Show that S=$\frac{375}{\pi r^{2}}+4 r(1+\pi)$.
S=$\frac{375}{{\pi}r^2}$+ar(b+$\pi$);a= ,b= .
4. Find $\frac{\mathrm{d} S}{\mathrm{~d} r}$=-$\frac{a}{{\pi}r^3}$+4(b+$\pi$);a= ,b= .
The greenhouse is designed so that the length of the steel used in the frame is minimized.
5. Show that the value of r for which S is a minimum is 2.43 $\mathrm{~m}$ , correct to 3 significant figures
6. Calculate the value of l for which S is a minimum.
7. Calculate the minimum value of S ≈ m.
