Exercise 37 Page 738

We are told that the surface area of a sphere is 100π mm^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 100π for S.A. in the formula and solve for r.

S.A.=4π r^2

Substitute

S.A.= 100π

100π=4π r^2

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Solve for r

DivEqn

.LHS /4π.=.RHS /4π.

100/4=r^2

RearrangeEqn

Rearrange equation

r^2=100/4

CalcQuot

Calculate quotient

r^2=25

RadicalEqn

sqrt(LHS)=sqrt(RHS)

r=sqrt(25)

CalcRoot

Calculate root

r=5

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 100π mm^2 is 5mm.

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 5, we can substitute its value for r into this formula.

V=4/3π r^3

Substitute

r= 5

V=4/3π ( 5)^3

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Simplify right-hand side

CalcPow

Calculate power

V=4/3π (125)

CommutativePropMult

Commutative Property of Multiplication

V=4/3(125)π

MoveRightFacToNum

a/c* b = a* b/c

V=500/3π

The volume of a sphere in terms of π with surface area 100π mm^2 is 5003π mm^3.