Exercise 28 Page 860

a We are given the following cone with a radius of

r = 6

centimeters and a slant height of

ℓ = 12

centimeters.

We are asked how doubling the slant height and the radius of the cone will affect the surface area of the cone. First, let's find the surface area of the given cone using the formula.

S_{1} = π r^{2} + π r ℓ

▼

Substitute values and evaluate

SubstituteII

$r = 6$ , $ℓ = 12$

S_{1} = π (6)^{2} + π (6) (12)

CalcPowProd

Calculate power and product

S_{1} = 36 π + 72 π

AddTerms

Add terms

S_{1} = 108 π

Therefore, the surface area of the initial cone is

108 π

square centimeters. Now, let's find the surface area when we double the radius and the slant height

S_{2} .

The new dimensions are

r = 2 \cdot 6 = 12

cm and

ℓ = 2 \cdot 12 = 24

cm.

S_{2} = π r^{2} + π r ℓ

▼

Substitute values and evaluate

SubstituteII

$r = 12$ , $ℓ = 24$

S_{2} = π (12)^{2} + π (12) (24)

CalcPowProd

Calculate power and product

S_{2} = 144 π + 288 π

AddTerms

Add terms

S_{2} = 432 π

This tells us that the surface area when we double its dimensions is

432 π

square centimeters. Now, let's divide

S_{2}

S_{1}

to find the change.

\frac{S _{2}}{S _{1}} = \frac{4 3 2 π}{1 0 8 π} = 4

Finally, we find that the surface area is multiplied by

4

when we double the radius and the slant height of the cone.

b From Part A we know that the surface area of the initial cone is

S_{1} = 108 π

square centimeters. We are asked how dividing the slant height and the radius of the cone by three will affect the surface area of the cone

S_{3} .

This tells us that new dimensions are

r = \frac{6}{3} = 2

cm and

ℓ = \frac{1 2}{3} = 4

cm.

S_{3} = π r^{2} + π r ℓ

▼

Substitute values and evaluate

SubstituteII

$r = 2$ , $ℓ = 4$

S_{2} = π (2)^{2} + π (2) (4)

CalcPowProd

Calculate power and product

S_{2} = 4 π + 8 π

AddTerms

Add terms

S_{2} = 12 π

The surface area when we divide its dimensions by three is

12 π

square centimeters. Now, let's divide

S_{3}

S_{1}

to find the change.

\frac{S _{3}}{S _{1}} = \frac{1 2 π}{1 0 8 π} = \frac{1}{9}

Finally, we find that the surface area is divided by

9

when we divide the radius and the slant height of the cone by three.

Subchapter links

Check Your Understanding p.859

Practice and Problem Solving p.859-861

H.O.T. Problems p.861

Standardized Test Practice p.862

Spiral Review p.862

Skills Review p.862