   Chapter 9.2, Problem 28E 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

For a regular square pyramid, the slant height of each lateral face has a measure equal to that of each edge of the base. If the lateral area is 200 i n 2 , find the volume of the pyramid. To determine

To find:

The volume of the pyramid.

Explanation

Given:

A regular square pyramid with slant height of each lateral face equal to that of each edge of the base i.e. l=x and the lateral area is 200in2, as below,

Properties Used:

A pyramid is made by connecting a base to an apex. The base is flat with straight edges, no curves, hence, a polygon and all other faces are triangles.

A regular pyramid is a pyramid whose base is a regular polygon and whose lateral edges are all congruent.

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.

The total surface area T of a pyramid with lateral area L and base area B is given by

T=L+B.

According to the Pythagorean theorem, in a right-angled triangle

hypotenuse2=base2+perpendicular2.

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.

The perimeter of the square is four times the side.

The volume V of a pyramid having a base area B and an altitude of length h is given by V=13Bh.

Calculation:

From the given figure

the sides of the square base measure x each and the apothem is half the base length. Hence, a=x2.

The slant height of each lateral face equal to that of each edge of the base i.e. l=x.

The square base perimeter P=4x

The lateral area is

L=12lP=12x4xL=2x2

But it is given the lateral area is 200in2

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