The Conical X-Factor

Geometry Level 3

Two right circular cones each share the same altitude and a height of h h . One cone has a base of radius R R and the other cone is inverted and has a base of radius r r as shown in the figure below.

The volume of the region common to both cones can be calculated by the formula:

V = π R 2 r 2 h X ( R + r ) 2 V =\frac {\pi R^2 r^2 h}{ X ( R + r )^2}

Find X X .


The answer is 3.

W Rose by W Rose , 63, USA

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2 solutions

Tom Engelsman
Nov 17, 2016

6 Helpful 0 Interesting 0 Brilliant 0 Confused
Dan Ley
Dec 29, 2016

By ratios, x r = h 2 h \frac{x}{r}=\frac{h_2}{h} and similarly x R = h 1 h \frac{x}{R}=\frac{h_1}{h}

⟹ x h R + x h r = h 1 + h 2 = h \implies \frac{xh}{R}+\frac{xh}{r}=h_1+h_2=h

Dividing by h h and cross-multiplying gives x = R r R + r x=\frac{Rr}{R+r}

The area of intersection is 1 3 π x 2 h 1 + 1 3 π x 2 h 2 = 1 3 π x 2 ( h 1 + h 2 ) = 1 3 π x 2 h \frac{1}{3}\pi x^2h_1 + \frac{1}{3}\pi x^2h_2= \frac{1}{3}\pi x^2(h_1+h_2)=\frac{1}{3}\pi x^2h

If we substitute in x 2 x^2 we get 1 3 π h R 2 r 2 ( R + r ) 2 = π R 2 r 2 h 3 ( R + r ) 2 ⟹ X = 3 \frac{1}{3}\pi h\frac{R^2r^2}{(R+r)^2}=\frac{\pi R^2r^2h}{3(R+r)^2} \implies X=3

3 Helpful 0 Interesting 0 Brilliant 0 Confused

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