2.66: Volume and related rates in a truncated cone

0606/23/M/J/20 — Question 11 · 7 marks

In this question all lengths are in centimetres.

The volume,

V

, of a cone of height

h

and base radius

r

is given by

V = \tfrac{1}{3}\pi r^{2}h

The diagram shows a large hollow cone from which a smaller cone of height

180

and base radius

90

has been removed. The remainder has been fitted with a circular base of radius

90

to form a container for water. The depth of water in the container is

w

and the surface of the water is a circle of radius

R

(a)

Find an expression for

R

in terms of

w

and show that the volume

V

of the water in the container is given by

V = \dfrac{\pi}{12}(w + 180)^{3} - 486000\pi

[3]

(b)

Water is poured into the container at a rate of

10000\text{ cm}^{3}\text{ s}^{-1}

. Find the rate at which the depth of the water is increasing when

w = 10

[4]