Chapter 9.CR, Problem 19CR

Elementary Geometry For College St...

7th Edition
Alexander + 2 others
ISBN: 9781337614085

Chapter
Section

Elementary Geometry For College St...

7th Edition
Alexander + 2 others
ISBN: 9781337614085
Textbook Problem

The solid shown consists of a hemisphere (half of a sphere), a cylinder, and a cone. Find the exact volume of the solid.

To determine

To find:

The exact volume of the solid.

Explanation

Formula used:

Volume of half sphere:

V=23Ï€r3

Volume of cylinder:

V=Ï€r2h

Volume of cone:

V=13Ï€r2h

Given:

The solid shown consists of a hemisphere (half of a sphere) a cylinder and a cone.

Calculation:

The volume of given solid is,

V=volumeÂ ofÂ halfÂ sphere+Â volumeÂ ofÂ cylinder+volumeÂ ofÂ coneV=23Ï€r3+Ï€r2h1+13Ï€r2h2Â

First we find the height of the cone,

Right circular cone as shown in the above figure.

Also, the slant height is the hypotenuse of a right circular cone with legs equal to the lengths of the altitude of the cone and the radius of the base.

Applying the Pythagorean Theorem,

l2=r2+h2252=32+h2225=9+h22InterchangingÂ theÂ valuesÂ andÂ subtractingÂ 9Â onÂ bothÂ sides,h22+9âˆ’9=25âˆ’9h22=16h2=16h2=4Â

The height of the cone is â€Šh2=4.

Next we find the height of the cylinder,

Already we know that, in the given solid the radius of cone is same as to the radius of half sphere.

Also, the radius of half sphere is same as to the height of the half sphere.

So, the height of the half sphere is â€Šh3=3

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