A bowl is shaped like a hemisphere with diameter 30 cm. A heavy ball with diameter 10 cm is placed in the bowl and water is poured into the bowl to a depth of h centimeters. Find the volume of water in the bowl.
Volume of water in the bowl
i) Volume when the ball is partially immersed
ii) Volume when ball is fully immersed
Volume of cap of sphere of radius and height is given by,
Hemisphere of diameter 30 cm
Heavy ball of diameter 10 cm
In order to calculate the volume of the water in the bowl, consider two cases- one is that the heavy ball is partially submerged and another is that the heavy ball is fully submerged.
Consider that water is filled upto height in the bowl.
The marked region shown above is the bowl filled with water on the right side of the heavy ball, if rotated about y axis, will get the left side of bowl filled with water at h since the bowl is symmetric about y axis. So, the best method is to use the washer method.
So, area is calculated as
To calculate outer radius
To calculate inner radius that is
Suppose that the ball is completely immersed in water.
To find the volume of the water in the bowl simply subtract the volume of the ball from the volume of the bowl.
Volume of ball is equal to
According to formula of volume of cap,
At height h, the water is filled which can be calculated by the equation