   Chapter 9.2, Problem 34E 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 foyer planned as an addition to an existing church is designed as a regular octagonal pyramid. Each side of the octagonal floor has a length of 10 f t , and its apothem measures 12 f t . If 800 f t 2 of plywood is needed to cover the exterior of the foyer (that is, the lateral area of the pyramid is 800 f t 2 ), what is the height of the foyer?

To determine

To find:

The height of the foyer.

Explanation

Given:

A foyer in shape of a regular octagonal pyramid and each side of the octagonal floor has a length of 10ft, and its apothem measures 12ft. The lateral area of the pyramid is 800ft2.

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.

Approach Used:

i) Calculate the perimeter of the base P.

ii) Substitute the values of P and the given lateral area in equation L=12lP to find the value of slant height l.

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

Calculation:

Since, foyer is in shape of a regular octagonal pyramid and each side of the octagonal floor has a length of 10ft.

Hence, the perimeter of the octagonal base is

P=8x=8×10=80ft

The lateral area of the pyramid is 800ft2

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