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Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621

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Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621
Textbook Problem

Find the volume of the described solid S.

A pyramid with height h and base an equilateral triangle with side a (a tetrahedron)

Chapter 5.2, Problem 52E, Find the volume of the described solid S. A pyramid with height h and base an equilateral triangle

To determine

To find:

The volume of a pyramid with height h and base an equilateral triangle with side a.

Explanation

1) Concept:

Definition of volume:

Let S be a solid that lies between x=a and  x=b. If the cross sectional area of S in the plane Px, through x and perpendicular to the x-axis, is  A(x), where A is a continuous function, then the volume of S is

V=limni=1nAxi*x=abAxdx 

2) Given:

Height of pyramid is h and basean equilateral triangle with side a.

3) Calculations:

Consider the triangle consisting of two vertices of the base and the center of the base.

This triangle is similar to the corresponding triangle at a height y,

So from the figure,

ab=ΑΒ Α=aΒb.

Also, by similar triangles,

bh=Βh-y Β=bh-yh.

These two equations imply that

Α=a(1-yh)

Since the cross section is an equilateral triangle, it has area

Ay=12·Α·32Α

After substituting the value of Α,

Ay=34·a2

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