   Chapter 9.2, Problem 35E 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 exhaust chute on a wood chipper has the shape of a frustum of a pyramid. With dimensions as indicated, find the volume (capacity) of the chipper’s exhaust chute. To determine

To find:

The volume (capacity) of the chipper’s exhaust chute.

Explanation

Given:

The exhaust chute on a wood chipper is in the shape of a frustum of a pyramid as shown 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.

A frustum of a pyramid is the portion of a pyramid that lies between its base and a parallel plane that intersects the pyramid.

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

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.

Approach Used:

i) Calculate the volume of the bigger pyramid VB.

ii) Calculate the volume of the smaller pyramid VS.

iii) Subtract the two to get the volume of the frustum VF=VBVS.

Calculation:

Calculating the volume of the bigger pyramid

with base as a square of side x=6in and height as h=32in.

The area of the square base is

B=x2=62B=36in2

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