Chapter 2.1, Problem 52PS

Solution

(a.)

An expression for the total volume of the three tennis balls in terms of $r$ .

It has been determined that, an expression for the total volume of the three tennis balls in terms of $r$ is $4\pi r^3$ .

Given:

A can of tennis balls consists of three spheres of radius $r$ stacked vertically inside a cylinder of radius $r$ and height $h$ .

Concept used:

The volume of a sphere of radius $r$ is given as $V=\frac{4}{3}\pi r^3$ .

Calculation:

It is given that the three tennis balls are spheres of radius $r$ each.

As discussed previously, the volume of a sphere of radius $r$ is given as $V=\frac{4}{3}\pi r^3$ .

Then, the volume of a tennis ball is $V=\frac{4}{3}\pi r^3$ .

Since the three tennis balls are identical, the total volume of the three tennis balls is $3V=4\pi r^3$ .

Conclusion:

It has been determined that, an expression for the total volume of the three tennis balls in terms of $r$ is $4\pi r^3$ .

(b.)

An expression for the volume of the cylinder in terms of $r$ and $h$ .

It has been determined that, an expression for the volume of the cylinder in terms of $r$ and $h$ is $\pi r^{2}h$ .

Given:

A can of tennis balls consists of three spheres of radius $r$ stacked vertically inside a cylinder of radius $r$ and height $h$ .

Concept used:

The volume of a cylinder of radius $r$ and height $h$ is given as $V=\pi r^{2}h$ .

Calculation:

It is given that the cylinder has radius $r$ and height $h$ .

As discussed previously, the volume of a cylinder of radius $r$ and height $h$ is given as $V=\pi r^{2}h$ .

Hence, this is the required expression.

Conclusion:

It has been determined that, an expression for the volume of the cylinder in terms of $r$ and $h$ is $\pi r^{2}h$ .

(c.)

An expression for $h$ in terms of $r$ .

It has been determined that, an expression for $h$ in terms of $r$ is $h=6r$ .

Given:

A can of tennis balls consists of three spheres of radius $r$ stacked vertically inside a cylinder of radius $r$ and height $h$ .

Concept used:

In the given situation, the height of the cylinder is the sum of the diameters of the three tennis balls.

Calculation:

It is given that the height of the cylinder is $h$ .

It is given that the radius of each of the tennis balls is $r$ .

Then, the diameter of each of the tennis balls is $2r$ .

So, the sum of the diameters of the three tennis balls is $2r+2r+2r=6r$ .

According to the given situation, $h=6r$ .

Hence, this is the required expression.

Conclusion:

It has been determined that, an expression for $h$ in terms of $r$ is $h=6r$ .

(d.)

The fraction of the can’s volume that is taken up by the tennis balls.

It has been determined that, the fraction of the can’s volume that is taken up by the tennis balls, is $\frac{2}{3}$ .

Given:

A can of tennis balls consists of three spheres of radius $r$ stacked vertically inside a cylinder of radius $r$ and height $h$ .

Concept used:

The fraction of the can’s volume that is taken up by the tennis balls can be determined by dividing the total volume of the three tennis balls by the volume of the cylindrical can.

Calculation:

As determined previously, the total volume of the three tennis balls is $4\pi r^3$ .

Similarly, as determined previously, the volume of the cylindrical can is $\pi r^{2}h$ .

Then, as discussed previously, the fraction of the can’s volume that is taken up by the tennis balls; as determined by dividing the total volume of the three tennis balls by the volume of the cylindrical can, is given as $\frac{4\pi r^3}{\pi r^{2}h}$ .

Simplifying,

  $\frac{4\pi r^3}{\pi r^{2}h}=\frac{4r}{h}$

Finally, as determined previously, $h=6r$ .

Put $h=6r$ in $\frac{4\pi r^3}{\pi r^{2}h}=\frac{4r}{h}$ to get,

  $\frac{4\pi r^3}{\pi r^{2}h}=\frac{4r}{6r}$

Simplifying,

  $\frac{4\pi r^3}{\pi r^{2}h}=\frac{2}{3}$

Thus, the fraction of the can’s volume that is taken up by the tennis balls is $\frac{2}{3}$ .

Conclusion:

It has been determined that, the fraction of the can’s volume that is taken up by the tennis balls, is $\frac{2}{3}$ .