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Minimise the cone circumscribed about a sphere

A cone of radius rr and height hh has volume 13πr2h\tfrac13\pi r^2h. A sphere of fixed radius aa and centre OO is inscribed in a right cone, touching its base and curved surface, as shown.

(i) Show that r2=a2hh−2ar^2=\dfrac{a^2h}{h-2a}. marks

(ii) Use differentiation to find the minimum volume of the cone in terms of aa, proving it is a minimum. marks

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Axial section of a cone containing an inscribed sphere. The cone has apex V, height h and base radius r; the sphere centre O is a above base centre C and its radius is a. The perpendicular OT to the right sloping side meets it at T; VO=h−a and VT is the square root of h squared minus 2ah.
Axial section of a cone containing an inscribed sphere. The cone has apex V, height h and base radius r; the sphere centre O is a above base centre C and its radius is a. The perpendicular OT to the right sloping side meets it at T; VO=h−a and VT is the square root of h squared minus 2ah.
Axial section of a cone containing an inscribed sphere. The cone has apex V, height h and base radius r; the sphere centre O is a above base centre C and its radius is a. The perpendicular OT to the right sloping side meets it at T; VO=h−a and VT is the square root of h squared minus 2ah.

[2021/ACJC/I/5]

CJC · Discussion · Question 12

Eliminate r using the tangent right triangle before differentiating with respect to h.

Read the worked solution