   Chapter 9.2, Problem 30E 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

Before the shingles of the steeple (see Exercise 29) are replaced, an exhaust fan is to be installed in the steeple. To determine what size exhaust fan should be installed, it is necessary to know the volume of air in the steeple. Find the volume of the regular square pyramid described in Exercise 29. To determine

To find:

The volume of the pyramid.

Explanation

Given:

A regular square pyramid as below,

and lateral edge measures 17ft and each base edge measures 16ft.

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 16ft each and the apothem is half the base length. Hence, a=162=8ft. The lateral edge measures 17ft or e=17ft.

The slant height

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