Analysis and Approaches for IBDP Maths Ebook 2 | Page 158

Your Practice Set – Analysis and Approaches for IBDP Mathematics
4. In the figure, a square-based prism of side length x and height y is inscribed in a right
circular cone of radius R and height H , where the four vertices of the top square face of
the prism touches the circular cone, and the centres of the square faces are vertically
below the vertex of the circular cone. It is also given that 0 �x� R and 0 � y� H .
Let V be the volume of the square-based prism.
(a)
By considering the distance between the centre of the top square face and one of
its vertices, show that
2H
y �H � x .
2R
(b) Hence, express V in terms of x , H and R .
2 2
(c) Show that V attains its maximum when x� R.
3
[6]
(d) Hence, find the maximum value of V , giving the answer in terms of H and R .
[3]
(e) Find the total surface area of the square-based prism when its volume attains its
maximum, giving the answer in terms of H and R .
[4]
[4]
[2]
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