Chapter 5.2, Problem 68E

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8th Edition

James Stewart

ISBN: 9781285740621

Textbook Problem

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.

To determine

**To find:**

Volume of water in the bowl

Explanation

**1) Formula:**

Volume of cap of sphere of radius

**2) Given:**

Hemisphere of diameter 30 cm

Heavy ball of diameter 10 cm

**3) Calculation:**

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.

Case 1:

Consider that water is filled upto height

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

Therefore,

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