# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2005 | Oct-Nov | (P1-9709/01) | Q#5

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Question

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  cm and height 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,  cm3, of the cylinder is given by

ii.       Given that  varies, find the stationary value of .

Solution

From the given information we can compile following data;

i.

In the given diagram;

The outer triangle (shaded red) and the inner triangle (top small triangle shaded black) are similar.

For similar triangles ratios of sides are equal;

To find the volume of cylinder;

Substituting expression for  derived above;

ii.

A stationary point  on the curve  is the point where gradient of the curve is equal to zero;

In the given case, ;

Therefore;

Therefore we find derivative of Area of Window;

Rule for differentiation of  is:

Rule for differentiation of  is:

Since at stationary point;

To find the value of volume of cylinder at stationary point, we substitute  in the expression of volume of cylinder derived in (i);