A racquetball with a diameter of 5.6 cm and a mass of 42 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 g and 0.36 cm3. How many pennies can be placed in the racquetball boat without sinking it? (The density of water is 1000 kg/m3.)
To determine the maximum number of pennies that can be placed in the racquetball boat without sinking it, we need to calculate the boat's buoyant force and compare it to the boat's weight.
Let's go step by step:
Step 1: Calculate the volume and density of the racquetball boat.
Since the racquetball is cut in half, the volume of the boat will be half of the original volume.
The volume of a sphere (racquetball) is given by the formula:
V = (4/3) * π * r^3
where r is the radius.
Given that the diameter of the racquetball is 5.6 cm, the radius (r) will be half the diameter:
r = 5.6 cm / 2 = 2.8 cm = 0.028 m
Using this radius, we can calculate the volume of the boat:
V_boat = (4/3) * π * r^3 / 2
Step 2: Calculate the mass of the boat.
Since the racquetball is cut in half, the mass of the boat will also be half of the original mass:
m_boat = 42 g / 2 = 21 g
Step 3: Calculate the density of the boat.
Density is given by the formula:
density = mass / volume
We can calculate the density of the boat:
density_boat = m_boat / V_boat
Step 4: Calculate the buoyant force.
Buoyant force is given by the formula:
buoyant force = volume of the fluid displaced * density of the fluid * acceleration due to gravity
Since the boat is floating in water, the fluid being displaced is water, and the density of water is given as 1000 kg/m^3:
buoyant force = V_boat * density_water * g
where g is the acceleration due to gravity, approximately 9.8 m/s^2.
Step 5: Calculate the weight of the boat.
The weight is given by:
weight = mass_boat * g
Step 6: Determine the maximum number of pennies.
To know the maximum number of pennies, we need to compare the buoyant force to the weight.
If the buoyant force is greater than or equal to the weight, the boat will not sink.
Since the density of the pennies is not provided, we can assume it is similar to the density of iron, which is approximately 7870 kg/m^3.
Dividing the density of the pennies by the density of water will give us the ratio of the densities, which we can then multiply by the boat's volume to find the maximum volume that can be filled with pennies.
max_volume_pennies = V_boat * (density_pennies / density_water)
Knowing the volume of a single penny (0.36 cm^3), we can calculate the maximum number of pennies:
max_number_pennies = max_volume_pennies / volume_penny
Finally, we have the maximum number of pennies that can be placed in the racquetball boat without sinking it.
To determine how many pennies can be placed in the racquetball boat without sinking it, we need to compare the buoyant force exerted by the water on the boat with the weight of the boat and the pennies.
Here are the steps to solve the problem:
Step 1: Calculate the volume of the racquetball boat:
The racquetball has a diameter of 5.6 cm, which gives a radius (r) of 2.8 cm (or 0.028 m).
The volume of a sphere is given by the formula: V = (4/3) * π * r^3
Substituting the values: V = (4/3) * 3.14 * (0.028^3)
V ≈ 0.0104 m^3
Step 2: Calculate the total mass of the boat:
The racquetball weighs 42 g, and we only need a half, so the mass of the boat (m_boat) is 21 g (or 0.021 kg).
The mass of one penny is 2.5 g, and the number of pennies we need to find (n_pennies).
The total mass of the boat with the pennies is: m_boat + n_pennies * 2.5 g
Step 3: Calculate the weight of the boat:
The weight of the boat is given by the formula: W_boat = m_boat * g
where g is the acceleration due to gravity (9.8 m/s^2)
Substituting the values: W_boat = 0.021 kg * 9.8 m/s^2
W_boat ≈ 0.206 N
Step 4: Calculate the buoyant force on the boat:
The buoyant force is given by the formula: F_buoyant = ρ_water * V_boat * g
where ρ_water is the density of water (1000 kg/m^3), V_boat is the volume of the boat (0.0104 m^3), and g is the acceleration due to gravity (9.8 m/s^2)
Substituting the values: F_buoyant = 1000 kg/m^3 * 0.0104 m^3 * 9.8 m/s^2
F_buoyant ≈ 101.92 N
Step 5: Set up and solve the equation:
To find the number of pennies that can be placed in the boat without sinking it, we need to determine when the buoyant force is greater than or equal to the weight of the boat.
F_buoyant ≥ W_boat
101.92 N ≥ 0.206 N + n_pennies * 2.5 N
Simplifying the equation:
101.92 N ≥ 0.206 N + 2.5 N * n_pennies
101.92 N - 0.206 N ≥ 2.5 N * n_pennies
101.714 N ≥ 2.5 N * n_pennies
Divide both sides by 2.5 N:
40.6856 ≥ n_pennies
Therefore, you can place a maximum of 40 pennies in the racquetball boat without sinking it.