Exercise 24 Page 646

We will begin by drawing a cone with radius b and height H. Then, we will cut it with a plane parallel to the base at a height of h and remove the upper part. That way, we obtain a frustum. To find the volume of the frustum, we subtract the volume of the smaller cone (the top) from the volume of the entire cone. V_(frustum) = V_(cone) - V_(top)The volume of the entire cone is one-third the area of the base multiplied by the height. V_(cone) = 1/3π b^2H The radius of the base of the smaller cone is a and its height is k. With this information we can find its volume. V_(top) = 1/3π a^2k Now, let's substitute the two volumes into the initial formula to find the volume of the frustum. Notice that H=h+k.

V_(frustum) = V_(cone) - V_(top)

SubstituteII

V_(cone)= 1/3π b^2H, V_(top)= 1/3π a^2k

V_(frustum) = 1/3π b^2H - 1/3π a^2k

▼

Simplify right-hand side

FactorOut

Factor out 1/3π

V_(frustum) = 1/3π (b^2H - a^2k)

Substitute

H= h+k

V_(frustum) = 1/3π (b^2( h+k) - a^2k)

Multiply

Multiply

V_(frustum) = 1/3π (b^2h+b^2k - a^2k)

FactorOut

Factor out k

V_(frustum) = 1/3π (b^2h+(b^2-a^2)k)

FacDiffSquares

a^2-b^2=(a+b)(a-b)

V_(frustum) = 1/3π (b^2h + (b+a)(b-a)k)

Let's consider the original cone whose height is H=h+k, and let's label the vertices of the two triangles formed.

Notice that XY∥ VW and by the Corresponding Angles Theorem we have that ∠ X ≅ ∠ ZVW. Also, ∠ Y ≅ ∠ ZWV because they are right angles. Thus, the Angle-Angle (AA) Similarity Theorem tells us that △ XYZ ~ △ VWZ.

Let's rewrite the left-hand side equation.

b/a = h+k/k

MultEqn

LHS * ak=RHS* ak

ak* b/a = ak * h+k/k

▼

Simplify

CancelCommonFac

Cancel out common factors

bk = a(h+k)

Multiply

Multiply

bk = ah+ak

SubEqn

LHS-ak=RHS-ak

bk-ak = ah

FactorOut

Factor out k

(b-a)k = ah

Finally, let's substitute the latter equation into the formula to find the volume of the frustum.

V_(frustum) = 1/3π (b^2h + (b+a)(b-a)k)

Substitute

(b-a)k= ah

V_(frustum) = 1/3π (b^2h + (b+a) ah)

▼

Simplify right-hand side

FactorOut

Factor out h

V_(frustum) = 1/3π h (b^2 + (b+a)a)

Multiply

Multiply

V_(frustum) = 1/3π h (b^2+ab+a^2)

CommutativePropAdd

Commutative Property of Addition

V_(frustum) = 1/3π h (a^2+ab+b^2)

We have written a formula to find the volume of a frustum in terms of a, b, and h.