Exercise 17 Page 431

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

Solid	Volume
Cylinder	π r^2 h
Cube	s^3

Let's try to rewrite the formula for calculating the volume of a cylinder so that it can be compared to the formula for the volume of a cube. First, we know the radius r is equal to half of the diameter d. We also know that the diameter of our cylinder has the same measurement as the side s of our cube. r=d/2=s/2 Additionally, the height h of our cylinder is also equal to the side of our cube. Using all of this we can rewrite the formula for the cylinder's volume.

V_\text{cylinder}=\pi r^2 h

SubstituteII

r= s/2, h= s

V_\text{cylinder}=\pi \bigg({\color{#0000FF}{\dfrac{s}{2}}}\bigg)^2 ({\color{#009600}{s}})

▼

Simplify right-hand side

CalcPow

Calculate power

V_\text{cylinder}=\pi \bigg(\dfrac{s^2}{4}\bigg) (s)

MoveRightFacToNum

a/c* b = a* b/c

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

MoveLeftFacToNum

a*b/c= a* b/c

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

MovePartNumRight

a* b/c=a/c* b

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

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

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

Substitute

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

V_\text{cylinder}=\dfrac{\pi}{4}({\color{#A800DD}{V_\text{cube}}})

CalcQuot

Calculate quotient

V_\text{cylinder}\approx0.785(V_\text{cube})

Now we can see that the volume of the cylinder is about 0.785 times the volume of the cube. This means the volume of the cylinder is about 78.5 % of the volume of the cube. Therefore, the volume of the cube is larger than the volume of the cylinder.