A solid, consisting of a right circular cone, standing on a hemis… - Vidyadip

Question

A solid, consisting of a right circular cone, standing on a hemisphere, is placed upright, in a right circular cylinder, full of water, and touches the bottom. Find the volume of water left in the cylinder, having given that the radius of the cylinder is 3 cm and its height is 6 cm; the radius of the hemisphere is 2 cm and the height of the cone is 4 cm. Give your answer to the nearest cubic centimeter.

✓

Answer

Radius of cylinder $= 3 cm$
Height of cylinder $= 6 cm$
Radius of hemisphere $= 2 cm$
Height of cone $= 4 cm$
Volume of water in the cylinder when it is full $=$
$\pi r ^2 h =\pi \times 3 \times 3 \times 6=54 \pi cm ^3$
Volume of water displaced = volume of cone + volume ofhemisphere
$=\frac{1}{3} \pi r ^2 h +\frac{2}{3} \pi r ^3$
$=\frac{1}{3} \pi r ^2( h +2 r )$
$=\frac{1}{3} \pi \times 2 \times 2(4+2 \times 2)$
$=\frac{1}{3} \pi \times 4 \times 8$
$=\frac{32}{3} \pi cm ^3$
Therefore, volume of water which is left
$=54 \pi-\frac{32}{3} \pi$
$=\frac{130}{3} \pi cm ^3$
$=\frac{130}{3} \times \frac{22}{7} cm ^3$
$=\frac{2860}{21} cm ^3$
$=136.19 cm ^3$
$=136 cm ^3$

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