Difference between revisions of "1960 IMO Problems/Problem 6"

Revision as of 22:28, 18 July 2016

Problem

Consider a cone of revolution with an inscribed sphere tangent to the base of the cone. A cylinder is circumscribed about this sphere so that one of its bases lies in the base of the cone. Let $V_1$ be the volume of the cone and $V_2$ be the volume of the cylinder.

a) Prove that $V_1 \neq V_2$ ;

b) Find the smallest number $k$ for which $V_1 = kV_2$ ; for this case, construct the angle subtended by a diamter of the base of the cone at the vertex of the cone.

Solution

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Part (a):

Let $R$ denote the radius of the cone, and let $r$ denote the radius of the cylinder and sphere. Let $l$ denote the slant height of the cone, and let $h$ denote the height of the cone.

Consider a plane that contains the axis of the cone. This plane will slice the cone and sphere into a circle $\omega$ inscribed in an isosceles triangle $T$ .

The area of $T$ may be computed in two different ways: $[T] = \frac{1}{2} \times r \times (2l + 2R) = r(l+R)$ $[T] = \frac{1}{2} \times 2R \times h = Rh$ From this, we deduce that $r = \frac{Rh}{l+R}$ .

Now, we calculate our volumes: $V_1 = \frac{1}{3}\pi R^2 h$ $V_2 = \pi r^2 \times 2r = 2\pi r^3 = \frac{2R^3 h^3}{(l+R)^3}$ Now, we will compute the quantity $\frac{3(l+R)^3}{\pi R^5 h} (V_1 - V_2)$ and prove that it is always greater than $0$ . Let $x = \frac{h}{R}$ . Clearly, $x$ can be any positive real number. Define $W_1 = \frac{3(l+R)^3}{\pi R^5 h} V_1$ and $W_2 = \frac{3(l+R)^3}{\pi R^5 h} V_2$ . We will calculate $W_1$ and $W_2$ in terms of $x$ and then compute the desired quantity $W_1 - W_2$ .

It is easy to see that: $W_1 = (\sqrt{x^2+1} + 1)^3$ $W_2 = 6x^2$

Now, let $u = \sqrt{x^2+1}$ . Since $x > 0$ , it follows that $u > 1$ . We now have: $W_1 = (u + 1)^3$ $W_2 = 6(u^2 - 1)$

Define $f(u) = W_1 - W_2$ . It follows that: $f(u) = (u+1)^3 - 6(u^2 - 1)$ $f(u) = u^3 - 3u^2 + 3u + 7$ $f(u) = (u-1)^3 + 8 > 8 > 0$

We see that $f(u) > 8$ for all allowed values of $u$ . Thus, $V_1 - V_2 > \frac{8\pi R^5 h}{3(l+R)^3}$ , meaning that $V_1 > V_2$ . We have thus proved that $V_1 \ne V_2$ , as desired.

Part (b):

From our earlier work in calculating the volumes $V_1$ and $V_2$ , we easily see that: $\frac{V_1}{V_2} = \frac{(u+1)^3}{6(u^2 - 1)}$ Re-expressing and simplifying, we have: $\frac{6V_1}{V_2} = \frac{(u+1)^2}{u-1} = (u-1) + 4 + \frac{4}{u-1}$ By the AM-GM Inequality, $(u-1) + \frac{4}{u-1} \ge 4$ , meaning that $\frac{V_1}{V_2} \ge \frac{4}{3}$ . Equality holds if and only if $u-1 = \frac{4}{u-1}$ , meaning that $u=3$ and $x = 2\sqrt{2}$ .

If we check the case $x = \frac{h}{R} = 2\sqrt{2}$ , we may calculate $V_1$ and $V_2$ : $V_1 = \frac{1}{3} \pi R^2 h = \frac{2\sqrt{2}}{3}\pi R^3$ $V_2 = 2\pi r^3 = 2\pi \left(\frac{R\times 2\sqrt{2} R}{4R}\right)^3 = \frac{\pi}{\sqrt{2}} R^3$ Indeed, we have $\frac{V_1}{V_2} = \frac{4}{3}$ , meaning that our minimum of $k = \frac{4}{3}$ can be achieved.

Thus, we have proved that the minimum value of $k$ such that $V_1 = kV_2$ is $\frac{4}{3}$ .

Now, let $\theta$ be the angle subtended by a diameter of the base of the cone at the vertex of the cone. We have the following: $\tan\left(\frac{\theta}{2}\right) = \frac{1}{x} = \frac{\sqrt{2}}{4}$ From the double-angle formula for tangent, $\theta = \arctan\left(\frac{4\sqrt{2}}{7}\right)$ This angle is easy to construct. Simply take any segment and treat it as a unit segment. Create a right triangle with legs of lengths $4\sqrt{2}$ and $7$ . This is straightforward, and the angle opposite the leg of length $4\sqrt{2}$ will be the desired angle $\theta$ .