An open barge has the dimensions l=22 m, w=40 m, and h=12m. If the barge is made out of 4.0 cm thick steel plate on each of its four sides and its bottom, what mass of coal can the barge carry in fresh water without sinking? Is there enough room in the barge to hold this amount of coal? ( the density of coal is about 1500 kg/m^3)
show me the solution.
To determine the mass of coal the barge can carry without sinking, we need to calculate the maximum weight the barge can support, considering its volume and the weight of the steel plates.
1. Calculate the volume of the barge:
Volume = length x width x height
= 22m x 40m x 12m
= 10,560 m^3
2. Calculate the total volume of the steel plates:
The barge has 4 sides and a bottom, each made of 4.0 cm thick steel plate.
Total volume of steel plates = (length x height of each side) + (width x height of each side) + (length x width x thickness of the bottom)
= (22m x 12m) + (40m x 12m) + (22m x 40m x 0.04m)
= 264m^2 + 480m^2 + 35.2m^2
= 779.2m^2
3. Calculate the volume of the open space in the barge:
Volume of open space = Volume of barge - Total volume of steel plates
= 10,560m^3 - 779.2m^3
= 9,780.8m^3
4. Calculate the mass of coal the barge can carry without sinking:
Mass of coal = Volume of open space x Density of coal
= 9,780.8m^3 x 1500 kg/m^3
= 14,671,200 kg
Therefore, the barge can carry up to 14,671,200 kg (or 14,671.2 metric tons) of coal without sinking.
To determine if there is enough room for this amount of coal, compare the calculated mass with the density of coal.
The density of coal is about 1500 kg/m^3,
which means 1 m^3 of coal weighs 1500 kg.
Comparing with the available volume of the open space in the barge (9,780.8 m^3), we can calculate the maximum mass the barge can carry based on the available volume.
Maximum mass of coal based on available volume = Volume of open space x Density of coal
= 9,780.8 m^3 x 1500 kg/m^3
= 14,671,200 kg (or 14,671.2 metric tons)
Since the calculated mass (14,671,200 kg) matches with the maximum mass based on available volume, there is enough room in the barge to hold this amount of coal.
To determine the mass of coal the barge can carry without sinking, we need to consider the weight of the barge itself and the weight of the coal, as well as the buoyant force that supports the barge in water.
First, let's calculate the volume of the barge. The barge is shaped like a rectangular box, so its volume can be determined by multiplying its length (l), width (w), and height (h):
Volume of barge = l * w * h
Plugging in the given values:
Volume of barge = 22 m * 40 m * 12 m
Volume of barge = 10560 m^3
Since the barge is made of steel plate, we need to consider the volume occupied by the steel as well. The steel plates are 4.0 cm thick on each of its four sides and the bottom. Thus, we need to subtract the volume occupied by the steel plates from the total volume of the barge to find the available volume for coal:
Volume occupied by steel = (l + w + l + w + h + h) * thickness of steel
Volume occupied by steel = (22 m + 40 m + 22 m + 40 m + 12 m + 12 m) * 0.04 m
Volume occupied by steel = 138.4 m * 0.04 m
Volume occupied by steel = 5.536 m^3
Available volume for coal = Volume of barge - Volume occupied by steel
Available volume for coal = 10560 m^3 - 5.536 m^3
Available volume for coal = 10554.464 m^3
Now, let's calculate the mass of coal that can be carried in this available volume. We know that the density of coal is 1500 kg/m^3, so:
Mass of coal = Volume of coal * Density of coal
Mass of coal = Available volume for coal * Density of coal
Mass of coal = 10554.464 m^3 * 1500 kg/m^3
Mass of coal = 15,831,696 kg
Therefore, the barge can carry approximately 15,831,696 kg (or 15,831.7 metric tons) of coal without sinking.
Now, let's check if there is enough room in the barge to hold this amount of coal. To do this, we compare the volume of the coal to the available volume for coal:
Volume of coal = Mass of coal / Density of coal
Volume of coal = 15,831,696 kg / 1500 kg/m^3
Volume of coal = 10,554.464 m^3
As we calculated earlier, the available volume for coal is approximately 10,554.464 m^3. Therefore, there is just enough room in the barge to hold the calculated mass of coal without sinking.
Remember that these calculations assume the barge is in fresh water. If the barge were in saltwater, we would need to consider the difference in buoyancy due to the higher density of saltwater.