Exercise 45 Page 854

Therefore, the diagonal of the base is d=6sqrt(2) inches. Now, let's analyze a cross section of the given cube and the sphere with a plane through D, B, H, and F.

Let's use the Pythagorean Theorem for △ BDF. Then we will solve the equation for r.

BD^2+BF^2=DF^2

SubstituteExpressions

Substitute expressions

6^2+(6sqrt(2))^2=(2r)^2

▼

Solve for r

PowProd

(a * b)^m=a^m* b^m

6^2+6^2(sqrt(2))^2=2^2r^2

PowSqrt

( sqrt(a) )^2 = a

6^2+6^2(2)=2^2r^2

CalcPowProd

Calculate power and product

36+72=4r^2

AddTerms

Add terms

108=4r^2

RearrangeEqn

Rearrange equation

4r^2=108

DivEqn

.LHS /4.=.RHS /4.

r^2=27

SqrtEqn

sqrt(LHS)=sqrt(RHS)

sqrt(r^2)=sqrt(27)

SqrtPowToNumber

sqrt(a^2)=a

r=sqrt(27)

SplitIntoFactors

Split into factors

r=sqrt(9* 3)

SqrtProd

sqrt(a* b)=sqrt(a)*sqrt(b)

r=sqrt(9)*sqrt(3)

CalcRoot

Calculate root

r=3sqrt(3)

Therefore, the radius of the sphere is r=3sqrt(3). Finally, let's find the volume of the sphere and round the answer to the nearest tenth.

V_\text{sphere}=\dfrac{4}{3}\pi r^3

▼

Substitute 3sqrt(3) for r and evaluate

Substitute

r= 3sqrt(3)

V_\text{sphere}=\dfrac{4}{3}\pi\left({\color{#0000FF}{3\sqrt{3}}}\right)^3

PowProd

(a * b)^m=a^m* b^m

V_\text{sphere}=\dfrac{4}{3}\pi (3)^3(\sqrt{3})^3

CalcPow

Calculate power

V_\text{sphere}=\dfrac{4}{3}\pi (27)\sqrt{27}

SplitIntoFactors

Split into factors

V_\text{sphere}=\dfrac{4}{3}\pi (27)\sqrt{9\cdot 3}

SqrtProd

sqrt(a* b)=sqrt(a)*sqrt(b)

V_\text{sphere}=\dfrac{4}{3}\pi (27)\sqrt{9}\cdot\sqrt{3}

CalcRoot

Calculate root

V_\text{sphere}=\dfrac{4}{3}\pi (27)(3)\sqrt{3}

Multiply

V_\text{sphere}=\dfrac{4}{3}\cdot 81\pi\sqrt{3}

MoveRightFacToNum

a/c* b = a* b/c

V_\text{sphere}=\dfrac{4\cdot 81\pi\sqrt{3}}{3}

Multiply

V_\text{sphere}=\dfrac{324\pi\sqrt{3}}{3}

CalcQuot

Calculate quotient

V_\text{sphere}=108\pi\sqrt{3}

▼

Round to 1 decimal place(s)

UseCalc

Use a calculator

V_\text{sphere}=587.670994\ldots

RoundDec

Round to 1 decimal place(s)

V_\text{sphere}\approx 587.7

This tells us that the volume of the sphere is 108πsqrt(3) cubic inches, which is about 587.7 cubic inches.

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