3.18: Sector formed into a cone; related rates

0606/21/M/J/25 — Question 9 · 12 marks

In this question, all lengths are in centimetres and all angles are in radians.

The diagram shows a sector

AOB

of a circle, centre

O

, radius

15

Angle

AOB = \dfrac{6\pi}{5}

The sector is made into a cone with points

A

and

B

touching, as shown.

(a)

Find the curved surface area of the cone.

[2]

(b)

Find the circumference of the circular top.

[2]

(c)

Hence find the radius of the circular top and the perpendicular height of the cone.

[2]

(d)(i)

Water is poured into the cone. When the depth of the water in the cone is

h

, the radius of the circular top of the water is

r

Find an expression for

r

in terms of

h

[1]

(d)(ii)

The water is poured into the cone at a constant rate of

27\text{ cm}^{3}

per second.

Find the rate at which the depth of the water is rising when the depth of the water is

4

[5]