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Glencoe Math: Course 3, Volume 2

GM

Glencoe Math: Course 3, Volume 2 View details

6. Changes in Dimensions

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Exercise 10 Page 646

Practice makes perfect

A frustum is the solid left after a cone is cut by a plane parallel to its base and the top cone is removed. We can see a frustum in the following graph. We want to decide if the smaller cone that is removed is similar to the original one.

Cone

To answer this question, we should check if their lengths — the radii of the bases and the heights — are proportional. We can read these lengths from the graph.

Top Cone height : 3 in . radius : 1.5 in . Original Cone height : 6 in . radius : 3 in .

We see that the dimensions of the original cone are twice the dimensions of the top cone.

Top Cone height : 3 in . radius : 1.5 in . Original Cone height : 2 (3) = 6 in . radius : 2 (1.5) = 3 in .

Since the lengths are proportional, the cones are similar.

We are asked to find the volumes of the top cone and the original cone.

Cone

For that, let's remember the formula for the volume of a cone. The volume

V

of a cone with radius

r

and height

h

can be written as follows.

V = \frac{1}{3} π r^{2} h

We can now substitute the dimensions of each cone into the formula and evaluate.

Top Cone height : 3 in . radius : 1.5 in . Original Cone height : 6 in . radius : 3 in .

Let's start with the top cone.

V = \frac{1}{3} π r^{2} h

$r = 1.5$ , $h = 3$

V = \frac{1}{3} π (1.5)^{2} (3)

Calculate power

V = \frac{1}{3} π (2.25) (3)

Multiply

V = \frac{1}{3} π (6.75)

CommutativePropMult

Commutative Property of Multiplication

V = \frac{1}{3} \cdot 6.75 \cdot π

MoveRightFacToNumOne

$\frac{1}{b} \cdot a = \frac{a}{b}$

V = \frac{6 . 7 5}{3} \cdot π

Calculate quotient

V = 2.25 π

Use a calculator

V = 7.068583 \dots

Round to $2$ decimal place(s)

V \approx 7.07

The volume of the top cone is about

7.07 in .^{3} .

Let's now find the volume of the original cone.

V = \frac{1}{3} π r^{2} h

$r = 3$ , $h = 6$

V = \frac{1}{3} π (3)^{2} (6)

Calculate power

V = \frac{1}{3} π (9) (6)

Multiply

V = \frac{1}{3} π (54)

CommutativePropMult

Commutative Property of Multiplication

V = \frac{1}{3} \cdot 54 \cdot π

MoveRightFacToNumOne

$\frac{1}{b} \cdot a = \frac{a}{b}$

V = \frac{5 4}{3} \cdot π

Calculate quotient

V = 18 π

Use a calculator

V = 56.548667 \dots

Round to $2$ decimal place(s)

V \approx 56.55

We found that about

56.55 in .^{3}

is the volume of the original cone.

We want to find the ratio of the volume of the smaller cone

V_{1}

to the volume of the larger cone

V_{2} .

We can use the values found in Part B of the exercise.

V_{1} V_{2} = 7.07 in .^{3} = 56.55 in .^{3}

Calculating this ratio means dividing

V_{1}

by

V_{2} .

Let's do it then!

\frac{V _{1}}{V _{2}}

$V_{1} = 7.07$ , $V_{2} = 56.55$

\frac{7 . 0 7}{5 6 . 5 5}

Use a calculator

0.125022 \dots

Round to $3$ decimal place(s)

0.125

Write as a fraction

\frac{1}{8}

The ratio equals

\frac{1}{8},

or

1 : 8 .

Last, we want to find the volume of the frustum. Notice that the frustum is a difference of two cones. We also found the volume of each of the cones.

Frustum

Let's then subtract the volumes. This will give us the volume of the frustum.

V_{2} - V_{1}

$V_{1} = 7.07$ , $V_{2} = 56.55$

56.55 - 7.07

Subtract term

49.48

We found that about

49.48 in .^{3}

is the volume of the frustrum.

Subchapter links

Guided Practice p.644

Independent Practice p.645-646

H.O.T. Problems p.646

Standardized Test Practice p.646-648

Extra Practice p.647

Common Core Review p.648