Exercise 19 Page 747

We are given the volumes of two similar figures and the surface area of the smaller figure. We want to find the surface area of the larger figure. c|c Smaller Figure & Larger Figure [0.8em] V= 27m^3 & V= 64m^3 S.A.= 63m^2 & S.A.= ? To do so we will start by finding the scale factor of the solids. If the scale factor of two similar solids is a:b, then the ratio of their volumes is a^3:b^3. With this information and knowing the volume of both figures, we can find the scale factor.

a^3/b^3=64/27

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Solve for a/b

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

(a/b)^3=64/27

RadicalEqn

sqrt(LHS)=sqrt(RHS)

a/b=sqrt(64/27)

RootQuot

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

a/b=sqrt(64)/sqrt(27)

CalcRoot

Calculate root

a/b=4/3

FracToScale

\dfrac a b = a:b

a:b=4:3

The scale factor is 4:3. We can now square each number to obtain the ratio of the areas of the figures.

a/b=4/3

RaiseEqn

LHS^2=RHS^2

(a/b)^2=(4/3)^2

PowQuot

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

a^2/b^2=16/9

FracToScale

\dfrac a b = a:b

a^2:b^2=16:9

The ratio of the areas is 16:9. We know that the area of the smaller figure is 63m^2. If we let x be the area of the larger figure, the ratio of x to 63 is 16:9.

x/63=16/9

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Solve for x

MultEqn

LHS * 63=RHS* 63

x=16/9(63)

MoveRightFacToNum

a/c* b = a* b/c

x=16(63)/9

CalcQuotProd

Calculate quotient and product

x=112

The surface area of the larger figure is 112m^2.