Analysis and Approaches for IBDP Maths Ebook 2 | Page 157

3. In the figure, a right inverted circular cone of radius r is inscribed in a sphere with centre
O and fixed radius R . The perpendicular height of the cone is h , where 0 � r � R � h .
P is the centre of its circular base where its circumference touches the sphere.
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Let V be the volume of the circular cone.
(a) Express V in terms of � , r and R .
(b)
(c)
(d)
(e)
Hence, show that
2 2 2 2
d V � r(2R R � r � 2R �3 r )
�
dr 3 R � r
Show that V attains its maximum when
2 2
2 2
r � R.
3
Hence, find the value of k if the maximum value of V is
.
k
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3
� R .
Find the minimum capacity of the empty space inside the sphere, giving the
answer in terms of R .
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