Exercise 36 Page 738

We are told that the surface area of a sphere is 9π ft^2 and want to find its volume.

We first need to find its radius. To do so, we will use the formula for the surface area of a sphere. The surface area of a sphere is four times the product of π and the square of the radius. S.A.=4π r^2Let's substitute 9π for S.A. in the formula and solve for r.

S.A.=4π r^2

Substitute

S.A.= 9π

9π=4π r^2

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Solve for r

DivEqn

.LHS /4π.=.RHS /4π.

9/4=r^2

RearrangeEqn

Rearrange equation

r^2=9/4

RadicalEqn

sqrt(LHS)=sqrt(RHS)

r=sqrt(9/4)

SqrtQuot

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

r=sqrt(9)/sqrt(4)

CalcRoot

Calculate root

r=3/2

Note that, when solving the above equation, we kept the principal root. This is because r is the radius of a sphere and therefore must be positive. The radius of a sphere with surface area 9π ft^2 is 32ft.

Now, let's find the volume of the sphere. The volume of a sphere is four thirds the product of π and the cube of the radius. V=4/3π r^3 Since we already know that the radius is 32, we can substitute its value for r into this formula.

V=4/3π r^3

Substitute

r= 3/2

V=4/3π ( 3/2)^3

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Simplify right-hand side

PowQuot

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

V=4/3π (3^3/2^3)

CalcPow

Calculate power

V=4/3π (27/8)

MultFrac

Multiply fractions

V=108/24π

ReduceFrac

a/b=.a /12./.b /12.

V=9/2π

The volume of a sphere in terms of π with surface area 9π ft^2 is 92π ft^3.