   Chapter 9.2, Problem 38E 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

Each edge of a regular tetrahedron (see Exercise 37) has length e .a) Show that the altitude of the tetrahedron measures h = 2 3 e .b) Show that the volume of the tetrahedron is V = 2 12 e 3 .c) Find the volume of the tetrahedron if each side measures e = 4 i n . To determine

(a)

To show:

The altitude of the tetrahedron measures h=23e.

Explanation

Given:

Each edge of a regular tetrahedron has length e.as below,

Properties Used:

The lateral area L of a regular pyramid with slant height of length l and perimeter P of the base is given by

L=12lP.

In a regular pyramid, the lengths of the apothem a of the base, the altitude h, and the slant height l satisfy the Pythagorean Theorem; that is, l2=a2+h2.

According to the Pythagorean theorem, in a right-angled triangle hypotenuse2=base2+perpendicular2.

Area of a right-angled triangle is 12×base×height.

Approach Used:

i) Using the property l2=a2+h2, find the height or altitude h by substituting the values of l and a.

ii) Substitute the values of base and height h in equation 12×base×height to get the area of each lateral face since it is a triangular pyramid.

iii) Compare the given value with the value from step (ii).

Calculation:

From the given figure

the sides of the square base and lateral edge all are e.

Considering the right-angled triangle with triangle base

the sides of the triangular base measure e each and the angle at the center for each base is =360°3=120° which is bisected by the perpendicular so the hypotenuses in the right-angled triangle is OA¯=side2sin60°

To determine

(b)

To show:

The volume of the tetrahedron is V=212e3.

To determine

(c)

To find:

The volume of the tetrahedron.

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