Exercise 32 Page 855

Since the diagonal of the cube is the diameter of the sphere, we will find the length of the diagonal of the cube and then divide this length by 2 to find the radius of the sphere. Consider just the cube. Note that the diagonal of the cube, the diagonal of the base, and one of the sides create a right triangle.

We can use the Pythagorean Theorem for the marked triangle to find the length of the diagonal of the cube. However, before we do so, we need to find the length of the diagonal of the base. Note that the base of the cube is a square. If we divide a square with a diagonal, we get two 45^(∘)-45^(∘)-90^(∘) triangles. Therefore, the diagonal length is sqrt(2) times the side length. Diagonal of the base: sqrt(2) * 8=8sqrt(2) Knowing the length of the diagonal of the base of our cube, we can use the Pythagorean Theorem to find the length of the diagonal of the cube. Let's do it!

a^2+b^2=c^2

SubstituteII

a= 8sqrt(2), b= 8

( 8sqrt(2))^2+ 8^2=c^2

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Solve for c

PowProdII

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

8^2(sqrt(2))^2+8^2=c^2

PowSqrt

( sqrt(a) )^2 = a

8^2(2)+8^2=c^2

CalcPow

Calculate power

64(2)+64=c^2

Multiply

Multiply

128+64=c^2

AddTerms

Add terms

192=c^2

RearrangeEqn

Rearrange equation

c^2=192

SqrtEqn

sqrt(LHS)=sqrt(RHS)

c=sqrt(192)

SplitIntoFactors

Split into factors

c=sqrt(64* 3)

SqrtProd

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

c=sqrt(64)sqrt(3)

CalcRoot

Calculate root

c=8sqrt(3)

Note that since a negative side length does not make any sense, we only needed to consider the positive solution. Therefore, we have that the diagonal of the cube is 8sqrt(3) centimeters. Also the diameter of the sphere is 8sqrt(3) centimeters. Radius is half of the diameter, so we can use the obtained length to find it. r=8sqrt(3)/2=4sqrt(3) The radius of our sphere is 4sqrt(3) centimeters long.