   Chapter 9.4, Problem 29E 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

A right circular cone is inscribed in a sphere. If the slant height of the cone has a length equal to that of its diameter, find the length of thea) radius of the base of the cone.b) altitude of the cone. To determine

To find:

The radius of the base of the right circular cone, inscribed in a sphere.

Explanation

Approach:

A sphere is a three dimensional solid figure, which is made up of all the points in space, which lie at a common distance, called the radius, from a fixed point called the center of the sphere.

The surface area of a sphere S =4πr2, where r is the radius of the sphere.

Volume of a sphere V = 43πr3.

For right circular cone of base radius (r), slant height (l) and altitude (h),

Total surface area of the cone = πrl+r.

Curved or lateral surface area of the cone = πrl.

Volume of the cone = 13πr2h.

The Pythagorean theorem for a right angle triangle states that the square of the hypotenuse is equal to the sum of the squares of the other two sides. The longest side of the right angle triangle is called the hypotenuse.

Calculation:

Consider a right circular cone inscribed in a sphere. Slant height of the cone is equal to the diameter of its base.

For right circular cone of base radius (r), slant height (l) and altitude (h) and radius of sphere = (R).

Given that:

slant height (l) = diameter(d) = 2· radius(r)

By using Pythagorean Theorem,

h2+r2=l2

Putting value of l = 2r

h2+r2=2r2

h2+r2=4r2

h2=3r2

h=3r… (1)

h=x + R, x is the distance from the centre of the sphere to the centre of the circular base of the cone

To determine

To find:

The altitude of the base of the right circular cone, inscribed in a sphere.

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