Five spheres are kept in a cone in such a way that each sphere touch each other and also touch the lateral surface of the cone, this is due to increasing radius of the spheres starting from the vertex of the cone. The radius of the smallest sphere is 16 cm.

If the radius of the fifth (i.e., largest) sphere be 81 cm, then find the radius of the third (i.e., middlemost) sphere.

25 cm

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25 \sqrt{3} c m

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36 cm

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Solution

The correct option is C 36 cm
From the concept of similarity of triangles. All the five quadrilaterals viz, AOA', BOB', COC', DOD' and EOE' are similar.
From the figure(ii)

$\frac{r_{2} - r_{1}}{r_{2} + r_{1}} = \frac{r_{3} - r_{2}}{r_{3} + r_{2}} = \frac{r_{4} - r_{3}}{r_{4} + r_{3}} = \frac{r_{5} - r_{4}}{r_{5} + r_{4}} = K \Rightarrow \frac{r_{2}}{r_{1}} = \frac{r_{3}}{r_{2}} = \frac{r_{4}}{r_{3}} = \frac{r_{5}}{r_{4}} = K$
(By componendo and dividendo)

It means all the radii are in GP.
Therefore, $\frac{r_{5}}{r_{1}} = (K)^{4} = \frac{81}{16} = (\frac{3}{2})^{4}$
$\Rightarrow K = \frac{3}{2} ∴ r_{3} = r_{1} (K)^{2} r_{3} = r_{1} \times \frac{9}{4} = \frac{9 r_{1}}{4} = \frac{9}{4} \times 16 = 36 c m$

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