3.20: Minimum surface area of hemisphere–cylinder solid

0606/22/M/J/21 — Question 9 · 8 marks

All lengths are in centimetres. A solid is made from a hemisphere of radius

x

and a cylinder of radius

x

and height

y

. The volume is

500\,\mathrm{cm}^{3}

(a)

Find an expression for

y

in terms of

x

and show that the surface area

S

is given by

S = 3\pi x^{2} + \dfrac{1000}{x}

[4]

(b)

Given that

x

can vary and

S

has a minimum, find the value of

x

for which

S

is a minimum.

[4]