a large rectangular raft (density 650 kg/m^3) is floating on a lake. the surface

Question

a large rectangular raft (density 650 kg/m^3) is floating on a lake. the surface area of the raft is 8.2 m^2 and ist volume is 1.80 m^3. the density of lake water is 1000kg/m^3.

A) calculate the height h of the portion of the raft that is above the surrounding water.

B) calculate the magnitude of the buoyant force on the raft.

C) if the average mass of a person is 75 kg, calculate the maximum number of people that can be on the raft without the top of the raft sinking below the surface of the water. (assume that the people are evenly distibuted on the raft)

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Explanation / Answer

This problem entails displacement and density. .The formula for density is p=m/V where p = density, m = mass, and V = volume. We must first find the mass of the raft so we use the formula and plug in the numbers. 650kg/m^3 = m / 1.8m^3 so the raft's mass is 1170 kg We now need the height of the raft so we use the formula V = l * w * h where l = length, w = width, and h = height. 1.80m^3 = 8.2m^2 * h so height = 0.2195 meters or 21.95 cm .Now that we have the height of the raft, we must find out how far it sinks into the water. To do this, we must calculate the water's displacement. We know, thanks to Archimedes' that with displacement of water, the mass of the object in water, displaces an equal amount of water. Therefore, to find the Volume of water displaced, we must use the density formula and plug in the numbers. Since we know that there will be 1170kg of water displaced because that is the mass of the raft, we can use this as m or mass. 1000kg/m^3 = 1170kg / V V = 1.17m^3 Now we need to find the height of the water displaced. The water will have the same area as the raft wince water conforms to whatever is in or around it. Therefore the area of the water will be 8.2m^2 also. Using this, we plug it into the formula for V which is V = l * w * h, the same as before. 1.17m^3 = 8.2m^2 * h so therefore we get the height of the displaced water to = 0.1429m or 14.29cm. Subtract this from the original height of the raft and you get 0.0768m or 7.68cm. Therefore there is 7.68cm of the raft above the water. To figure out how many people could fit, you again use the density formula of p = m / V. In this case you would use the density of water of 1000kg/m^3, the mass of the displaced water + the mass of the people, which is the same as the mass of the raft if the raft is submerged, and the Volme of the displaced water which is the same as the mass of the raft if it is submerged. The total mass would be 1170kg (the mass of the raft) + the number of people times "x" because we do not know how many people there are. So the total mass would be 1170 + 75x. p = m / V 1000kg/m^3 = (1170kg + 75x) / 1.8m^3 1800kg = 1170kg + 75x 630kg = 75x x = 8.4 implies 8 people can

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