Exercise 24 Page 640

Let's analyze the given frustum of a square pyramid.

First, let's prolong the side edges that all of them intersect in one point.

Now, let's analyze the intersection of the big pyramid with a plane that goes through the height and is parallel to one of the base edges.

Let x denotes EF. Let's find CG in terms of a, b, and h. Since FE and AB are parallel, the triangles Δ ABC and Δ FEC are similar. As they are similar, the ratios between sides in the triangles are the same for each triangle.

CG/FE=CD/AB

SubstituteExpressions

Substitute expressions

x/b=x+h/a

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Solve for x

MultEqn

LHS * b=RHS* b

x=x+h/a* b

MultEqn

LHS * a=RHS* a

ax=b(x+h)

Distr

Distribute b

ax=bx+bh

SubEqn

LHS-bx=RHS-bx

ax-bx=bh

FactorOut

Factor out x

x(a-b)=bh

DivEqn

.LHS /(a-b).=.RHS /(a-b).

x=bh/a-b

Now, we will find the volumes of the smaller and the bigger pyramid. Then, by subtracting them, we will find the volume of the frustum.

Small Pyramid

The height of the smaller pyramid is x= bha-b. The base is a square of area b^2. Now, let's use the formula for the volume of a pyramid to find its volume, V_s.

V_s=1/3 A_(Base)* H

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Substitute values and evaluate

SubstituteValues

Substitute values

V_s=1/3 b^2* bh/a-b

MoveRightFacToNumOne

1/b* a = a/b

V_s=b^2/3*bh/a-b

MultFrac

Multiply fractions

V_s=b^3h/3(a-b)

Big Pyramid

Now, let's find the volume of the bigger pyramid, V_b. The area of its base is a^2. Let's find its height.

CD = h+x

Substitute

x= bh/a-b

CD = h+ bh/a-b

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Simplify right-hand side

NumberToFrac

a = (a-b)* a/(a-b)

CD=(a-b)h/a-b+bh/a-b

Distr

Distribute h

CD=ah-bh/a-b+bh/a-b

AddFrac

Add fractions

CD=ah-bh+bh/a-b

AddSubTerms

Add and subtract terms

CD = ah/a-b

Now, let's use the formula for the volume of a pyramid to find its volume, V_b.

V_b=1/3 Area of Base* Height

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Substitute values and evaluate

SubstituteValues

Substitute values

V_b=1/3 a^2* ah/a-b

MoveRightFacToNumOne

1/b* a = a/b

V_b=a^2/3*ah/a-b

MultFrac

Multiply fractions

V_b=a^3h/3(a-b)

Volume of Frustum

The volume of the smaller pyramid is V_s= b^3h3(a-b), and the volume of the bigger one is V_b= a^3h3(a-b). The volume of the frustum is the difference between the volumes of the pyramids. Let's find it.

V= V_b- V_s

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Substitute values and evaluate

SubstituteExpressions

Substitute expressions

V= a^3h/3(a-b)- b^3h/3(a-b)

SubFrac

Subtract fractions

V=a^3h-b^3h/3(a-b)

FactorOut

Factor out h

V=(a^3-b^3)h/3(a-b)