Math(s) Maximus

Calculus Level 5

The top of an inner sphere of radius $r$ is internally tangent at the top of a sphere of unit radius.

The inner cone's apex intersects the inner sphere's centroid while the bottom of the inner sphere is tangent to the center of the radius at the inner cone's bottom.

The maximum volume of the inner cone can be represented by $\pi \times \left( \dfrac AB \right)^4$ , where $A$ and $B$ are coprime positive integers. Find $A+B$ .

The answer is 5.

1 solution

Mark Hennings
Jan 25, 2017

The radius of the base of the cone is $x$ , where $x^2 + (1-2r)^2 = 1$ , so that $x^2 = 4r(1-r)$ . Thus the volume of the cone is $V \; = \; \tfrac13 \times \pi x^2 \times r \; = \; \tfrac43\pi r^2(1-r)$ and so $V'(r) \; = \; \tfrac43\pi r(2 - 3r)$ which shows that $V$ is maximised when $r=\tfrac23$ . Thus $V_{\mathrm{max}} \; =\; \tfrac43\pi \times \tfrac49 \times \tfrac13 \; = \; \pi \big(\tfrac23\big)^4$ making the answer $2+3=\boxed{5}$ .

Math(s) Maximus

The answer is 5.

1 solution

0 pending reports