Cambridge International AS and A Level Mathematics Pure Mathematics 1

Differentiation

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5

12  A garden is planned with a lawn area of 24 m^2 and a path around the edge.

The dimensions of the lawn and path are as shown in the diagram.

(i)  Write down an expression for y in terms of x.

(ii) Find an expression for the overall area of the garden, A, in terms of x.

(iii) Find the smallest possible overall area for the garden.

13  The diagram shows the cross-section of a hollow cone and a circular cylinder.

The cone has radius 6 cm and height 12 cm, and the cylinder has radius r cm

and height h cm. The cylinder just fits inside the cone with all of its upper

edge touching the surface of the cone.

(i)  Express h in terms of r and hence show that the volume, V cm^3 , of the

cylinder is given by

V = 12πr^2 – 2πr^3

(ii) Given that r varies, find the stationary value of V.

[Cambridge AS & A Level Mathematics 9709, Paper 1 Q5 November 2005]

1 m

y m

x m

1 m

1.5 m

1.5 m

lawn

K cm

U cm

6 cm

12 cm