Question

A racquetball with a diameter of \(5.6 \mathrm{~cm}\) and a mass of \(42 \mathrm{~g}\) is cut in half to make a boat for American pennies made after \(1982 .\) The mass and volume of an American penny made after 1982 are \(2.5 \mathrm{~g}\) and \(0.36 \mathrm{~cm}^{3}\). How many pennies can be placed in the racquetball boat without sinking it?

Short answer

Answer: The boat can hold 18 pennies without sinking.

Step-by-step solution

Find the volume of the racquetball

The racquetball is a sphere with a diameter of \(5.6 \mathrm{~cm}\). To find the volume of the racquetball, we can use the formula for the volume of a sphere, which is \(V = \frac{4}{3}\pi r^{3}\), where \(r\) is the radius of the sphere. The radius is half the diameter, so \(r = 2.8 \mathrm{~cm}\). Therefore, the volume of the racquetball is \(V = \frac{4}{3}\pi (2.8)^{3} = 91.79\mathrm{~cm}^3.\)

Find the volume of the boat

Since the racquetball is cut in half to make the boat, the volume of the boat is half the volume of the racquetball. So, the volume of the boat is \(V_{boat}=\frac{1}{2} \times 91.79\,\mathrm{~cm}^{3} =45.9\,\mathrm{~cm}^{3}.\)

Calculate the weight of the water that the boat can displace before sinking

The boat can hold water with the same weight as its volume. The density of water is \(1\,\mathrm{g/cm}^{3}\) so to find the weight of water, we multiply the volume of the boat by the density of water. Weight of water = \(45.9\,\mathrm{cm}^{3} \times 1\,\mathrm{g/cm^{3}} =45.9\,\mathrm{g}\).

Calculate the total mass that the boat can hold

The total mass that the boat can hold is the sum of the boat's mass and the weight of water it can displace before sinking. The mass of the racquetball is \(42\,\mathrm{g}\), so the mass of the boat (half of the racquetball) is \(21\,\mathrm{g}\). Therefore, the total mass that the boat can hold is \(45.9 + 21 = 66.9\,\mathrm{g}\).

Find the mass of pennies the boat can hold

The mass of pennies that the boat can hold is the total mass that the boat can hold minus the mass of the boat itself. So the mass of pennies the boat can hold is \(66.9\,\mathrm{g} - 21\,\mathrm{g}= 45.9\,\mathrm{g}\).

Calculate the number of pennies

To find the number of pennies, divide the mass of pennies the boat can hold by the mass of one penny, which is \(2.5\,\mathrm{g}\). Number of pennies = \(\frac{45.9}{2.5} = 18.36\). Since we cannot have a fraction of a penny, the boat can hold 18 pennies without sinking.