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A conical vessel of radius 6 cm and height 8 cm is completely filled with water. A metal sphere is now lowered into the water. The size of the sphere is such that when it touches the inner surface. It just gets immersed. What is the fraction of water the overflows from the conical vessel?

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\frac{3}{8}

\frac{5}{8}

\frac{7}{8}

\frac{5}{16}

answer is A.

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Detailed Solution

Given,
Radius of conical vessel = 6 cm
Height of conical vessel = 8 cm
Here, we can given the diagram for the problem as,
Question Image

By the property of similar triangles,

\frac{8 - r}{10} = \frac{r}{6}

\begin{matrix} 48 - 6 r = 10 r \\ r = 3 \end{matrix}

Now calculate the fraction of water overflows by comparing the volumes,

= \frac{Volume of sphere}{Volume of cone}

= \frac{\frac{4}{3} π (3)^{3}}{\frac{1}{3} π (6)^{2} (8)}

= \frac{\frac{4}{3} \times (\frac{22}{7}) \times 3 \times 3}{\frac{1}{3} \times (\frac{22}{7}) \times 6 \times 6 \times 8}

= \frac{3}{8}

This is the required fraction.
Option 1 is correct.

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