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Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621

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BuyFindarrow_forward

Calculus (MindTap Course List)

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 r and height h is given by,

Vcaph=πh2r-h3= 13πh2(3r-h)

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 y 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

A=πouter radius2-πinner radius2

To calculate outer radius

x2=152-(15-y)2

x=152-(15-y)2

To calculate inner radius that is r

x2=52-(5-y)2

x=52-(5-y)2

Therefore,

A(y)=π152-(15-y)22-π52-(5-y)22

A(y)=π[152-15-y2]-π[52-5-y2]

A(y)=π[225-(225-30y+y2)]-π[25-(25-10y+y2)]

A(y)=π[225-225+30y-y2]-π[25-25+10y-y2]

Ax

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