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

Revision as of 00:48, 7 July 2015

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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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 $\frac{3}{\pi R^3} (V_1 - V_2)$ and prove that it is always greater than $0$ . Let $x = \frac{h}{R}$ . Clearly, $x$ can be any positive real number.

It is easy to see that: $V_1 = \frac{\pi R^3}{3} (\sqrt{x^2+1} + 1)^3$ $V_2 = \frac{\pi R^3}{3} 6x^2$

Now, let $u = \sqrt{x^2+1}$ . Since $x > 0$ , it follows that $u > 1$ . We now have: $V_1 = \frac{\pi R^3}{3} (u + 1)^3$ $V_2 = \frac{\pi R^3}{3} 6(u^2 - 1)$

Define $f(u) = \frac{3}{\pi R^3} (V_1 - V_2)$ . 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, $V_1 - V_2 > \frac{8\pi}{3} R^3$ , meaning that $V_1 \ne V_2$ .

@@ Line 8: / Line 8: @@
 ==Solution==
 {{solution}}
+Let <math>R</math> denote the radius of the cone, and let <math>r</math> denote the radius of the cylinder and sphere. Let <math>l</math> denote the slant height of the cone, and let <math>h</math> 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 <math>\omega</math> inscribed in an isosceles triangle <math>T</math>.
+The area of <math>T</math> may be computed in two different ways:
+<cmath>[T] = \frac{1}{2} \times r \times (2l + 2R) = r(l+R)</cmath>
+<cmath>[T] = \frac{1}{2} \times 2R \times h = Rh</cmath>
+From this, we deduce that <math>r = \frac{Rh}{l+R}</math>.
+Now, we calculate our volumes:
+<cmath>V_1 = \frac{1}{3}\pi R^2 h</cmath>
+<cmath>V_2 = \pi r^2 \times 2r = 2\pi r^3 = \frac{2R^3 h^3}{(l+R)^3}</cmath>
+Now, we will compute the quantity <math>\frac{3}{\pi R^3} (V_1 - V_2)</math> and prove that it is always greater than <math>0</math>.
+Let <math>x = \frac{h}{R}</math>. Clearly, <math>x</math> can be any positive real number.
+It is easy to see that:
+<cmath>V_1 = \frac{\pi R^3}{3} (\sqrt{x^2+1} + 1)^3</cmath>
+<cmath>V_2 = \frac{\pi R^3}{3} 6x^2</cmath>
+Now, let <math>u = \sqrt{x^2+1}</math>. Since <math>x > 0</math>, it follows that <math>u > 1</math>. We now have:
+<cmath>V_1 = \frac{\pi R^3}{3} (u + 1)^3</cmath>
+<cmath>V_2 = \frac{\pi R^3}{3} 6(u^2 - 1)</cmath>
+Define <math>f(u) = \frac{3}{\pi R^3} (V_1 - V_2)</math>. It follows that:
+<cmath>f(u) = (u+1)^3 - 6(u^2 - 1)</cmath>
+<cmath>f(u) = u^3 - 3u^2 + 3u + 7</cmath>
+<cmath>f(u) = (u-1)^3 + 8 > 8 > 0</cmath>
+We see that <math>f(u) > 8</math> for all allowed values of <math>u</math>. Thus, <math>V_1 - V_2 > \frac{8\pi}{3} R^3</math>, meaning that <math>V_1 \ne V_2</math>.
 ==See Also==
 {{IMO7 box|year=1960|num-b=5|num-a=7}}

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

Revision as of 00:48, 7 July 2015

Problem

Solution

See Also