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Big Ideas Math: Modeling Real Life, Grade 8

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Big Ideas Math: Modeling Real Life, Grade 8 View details

Practice Test

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Exercise 10 Page 458

Compare the formulas for calculating the volume of both three-dimensional solids.

Cube, see solution.

We are asked to find which of two different three-dimensional solids has a greater volume without calculating the actual volumes.

The solids are a cube and a sphere. Let's look at the formulas for calculating the volume of each of these types of solids.

Solid	Volume
Cube	s^3
Sphere	4/3 π r^3

Let's try to rewrite the formula for calculating the volume of a sphere so that it can be compared to the formula for the volume of a cube. First, we know that the diameter d is equal to twice the radius r. d=2 r=18m This means the diameter of the sphere is the same measurement as the side s of the cube. In other words, the radius of the sphere is equal to half of the side of the cube. d= s ⇒ r=s/2 Now, we can rewrite the formula for the sphere's volume.

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

r= s/2

V_\text{sphere}=\dfrac{4}{3} \pi \bigg( \dfrac{{\color{#0000FF}{s}}}{2} \bigg)^3

▼

Simplify right-hand side

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

V_\text{sphere}=\dfrac{4}{3} \pi \dfrac{s^3}{2^3}

Calculate power

V_\text{sphere}=\dfrac{4}{3} \pi \dfrac{s^3}{8}

CommutativePropMult

Commutative Property of Multiplication

V_\text{sphere}=\dfrac{4}{3} \cdot \dfrac{s^3}{8}\pi

Multiply fractions

V_\text{sphere}=\dfrac{4\cdot s^3}{3\cdot 8}\pi

SplitIntoFactors

Split into factors

V_\text{sphere}=\dfrac{4\cdot s^3}{3\cdot 2\cdot 4}\pi

CancelCommonFac

Cancel out common factors

V_\text{sphere}=\dfrac{\cancel{{\color{#FF0000}{4}}}\cdot s^3}{3\cdot 2\cdot \cancel{{\color{#FF0000}{4}}}}\pi

Simplify quotient

V_\text{sphere}=\dfrac{s^3}{3\cdot 2}\pi

Multiply

V_\text{sphere}=\dfrac{s^3}{6}\pi

MoveRightFacToNum

a/c* b = a* b/c

V_\text{sphere}=\dfrac{\pi s^3}{6}

MovePartNumRight

a* b/c=a/c* b

V_\text{sphere}=\dfrac{\pi}{6} s^3

Since s^3 is the volume of the cube, we can rewrite the formula to compare the two volumes.

V_\text{sphere}=\dfrac{\pi}{6} s^3

s^3={\color{#0000FF}{{\color{#A800DD}{V_\text{cube}}}}}

V_\text{sphere}=\dfrac{\pi}{6} ({\color{#A800DD}{V_\text{cube}}})

Calculate quotient

V_\text{sphere}\approx0.52(V_\text{cube})

Now we can see that the volume of the sphere is about 0.52 times, or about 52 % of, the volume of the cube. Therefore, the volume of the cube is larger than the volume of the sphere.

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