1.9: Maximise a cylinder and reform it as a sphere

4PM1/2R/June/2022 — Question 11 · 12 marks

1.9 diagram 1
Figure 3 shows a solid metal right circular cylinder of radius r cm\displaystyle r\,\mathrm{cm} and height h cm\displaystyle h\,\mathrm{cm}.
The total surface area of the cylinder is 600 cm2\displaystyle 600\,\mathrm{cm}^2. The volume of the cylinder is V cm3\displaystyle V\,\mathrm{cm}^3.
(a) Show that
V=300r−πr3.V=300r-\pi r^3.
(4)
Given that r\displaystyle r can vary,
(b) (i) use calculus to show that the exact value of r\displaystyle r for which V\displaystyle V is a maximum is
r=100π,r=\sqrt{\frac{100}{\pi}},
(ii) justify that this value of r\displaystyle r gives a maximum value of V\displaystyle V.
(5)
The cylinder is melted down and reformed into a sphere of radius p cm\displaystyle p\,\mathrm{cm}.
(c) Find, to one decimal place, the greatest possible value of p\displaystyle p.
(3)