   Chapter 9.CR, Problem 19CR Elementary Geometry For College St...

7th Edition
Alexander + 2 others
ISBN: 9781337614085

Solutions

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=23πr3

Volume of cylinder:

V=πr2h

Volume of cone:

V=13πr2h

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=volume of half sphere+ volume of cylinder+volume of coneV=23πr3+πr2h1+13πr2h2

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+h22Interchanging the values and subtracting 9 on both sides,h22+99=259h22=16h2=16h2=4

The height of the cone is h2=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 h3=3

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