A hollow rubber ball with a diameter of 5.57 cm and a mass of 38.9 g is cut in half to make a boat (in a bathtub) 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 boat without sinking it?
I explained how to do this yesterday.
See
http://www.jiskha.com/display.cgi?id=1319564782
The solution utilizes Archimedes' Principle.
[(pi*Dball^3)/12]*Pwater = (1/2)*38.9g + Mp
where Pwater = 1.00 g/cm^3 is density of water and Mp is the mass of copper pennies, in grams.
45.24 g = 19.45 g + Mp
Mp = 25.79 g
Mp is a bit more than the mass of ten pennies. An eleventh would sink it.
thank you drwls !
To find out how many pennies can be placed in the boat without sinking it, we need to compare the mass of the pennies to the buoyant force of the boat.
First, let's calculate the volume of the hollow rubber ball boat. The volume of a sphere can be calculated using the formula:
V = (4/3) * π * r^3
where V is the volume and r is the radius of the sphere. Since the ball is cut in half, the radius of the boat will be half the original radius.
Given that the diameter of the rubber ball is 5.57 cm, the radius (r) will be half of that, which is 2.785 cm.
Using the formula, we can now calculate the volume of the boat:
V = (4/3) * 3.14 * (2.785 cm)^3
V ≈ 83.08 cm^3
Now that we have the volume of the boat, let's determine the maximum mass it can hold without sinking. We can use Archimedes' principle, which states that the buoyant force acting on a submerged object is equal to the weight of the fluid displaced by the object.
The buoyant force (FB) can be calculated as:
FB = ρ * g * V
where ρ is the density of the fluid and g is the acceleration due to gravity. For this problem, let's assume that the boat is placed in a bathtub filled with water, and hence ρ will be the density of water, which is approximately 1 g/cm^3.
Using the formula, we can calculate the buoyant force:
FB = 1 g/cm^3 * 9.8 m/s^2 * 83.08 cm^3
FB ≈ 815.24 g
Now, let's calculate the total mass of the boat (Mboat). Since the rubber ball was cut in half, the mass of the boat will be half of the original mass.
Mboat = 38.9 g / 2
Mboat = 19.45 g
To find out how many pennies can be placed in the boat without sinking it, we need to subtract the mass of the boat from the buoyant force and divide the result by the mass of one penny:
Number of pennies = (FB - Mboat) / Mass of one penny
Number of pennies = (815.24 g - 19.45 g) / 2.5 g
Number of pennies ≈ 318
Therefore, approximately 318 pennies can be placed in the boat without sinking it.