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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 rectangular base with dimensions b and 2b

To determine

To find:

The volume of a pyramid with height h and rectangular base with dimensions b and 2b.

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, base is b, and 2b.

3) Calculations:

Consider the pyramid having the apex point at the origin and its axis coinciding with the  x-axis.

From figure,

Consider the similar triangles, ABC & ADE,

For a cross-section at height h-x and base b,

Using the properties of similar triangles,

BCDE=ABBD

That is,

bx/2b/2=xh

Therefore, solving for bx,

bx=b·xh

Similarly,

For a cross-section at height h-x and base 2b,

lx=2bxh

Therefore, we have a rectangular cross section of width bx=bxh and length lx=2bxh  and having thickness

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