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McGraw Hill Integrated II, 2012

MH

McGraw Hill Integrated II, 2012 View details

8. Congruent and Similar Solids

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Exercise 12 Page 867

If the scale factor of two similar solids is a:b, then the ratio of their corresponding areas is a^2:b^2, and the ratio of their corresponding volumes is a^3:b^3.

125:8

Practice makes perfect

Similar solids have the same shape and all of their corresponding dimensions are proportional. The ratio of the corresponding linear dimensions is the scale factor. If the scale factor of two similar solids is a:b, then the ratio of their corresponding areas is a^2:b^2. Consider the given spheres.

Let's write the ratio of the surface areas as a fraction and take square roots to find the scale factor.

a^2/b^2=100π/16π

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Solve for a/b

a/b=.a /4π./.b /4π.

a^2/b^2=25/4

a^m/b^m=(a/b)^m

(a/b)^2=25/4

sqrt(LHS)=sqrt(RHS)

a/b=sqrt(25/4)

sqrt(a/b)=sqrt(a)/sqrt(b)

a/b=sqrt(25)/sqrt(4)

Calculate root

a/b=5/2

\dfrac a b = a:b

a:b=5:2

The scale factor is 5:2. We can now cube each number to obtain the ratio of the volumes of the spheres.

a/b = 5/2

LHS^3=RHS^3

( a/b )^3 = ( 5/2 )^3

(a/b)^m=a^m/b^m

a^3/b^3 = 5^3/2^3

Calculate power

a^3/b^3 = 125/8

\dfrac a b = a:b

a^3 : b^3 = 125 : 8

The ratio of the volumes of the spheres is 125:8.

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Exercises p.867-870