Buoyancy (A Boat in Water vs Alcohol)

[Rapier]

Homework Statement

You take a toy boat, float it in a tub of water and one half of the boat is below the surface. According to Archimedes’ Principle, how much of the boat would be below the surface if you tried to float it in a tub of alcohol?
→Less than half.
→Half.
→More than half, but still floating.
→All of it because it would sink.
→It depends on the depth of the tub.
→It depends on the length and width of the boat.

Homework Equations

ρ(water) = 1e3
ρ(alcohol) = .806e3

The Attempt at a Solution

I know that since the boat floats with half of it submerged in regular water that the volume of half of the boat is equal to the buoyant force. I know that alcohol is a lot less dense than water (about 20% less). I can narrow it down to either sinking or 'more than half but still floating.' I think it might sink because alcohol is not dense enough to support the boat, but I also think it might be possible for the difference in density between water and alcohol to be offset by the increased mass of displaced alcohol.

I was trying to use my equations to calculate the forces, but I'm not sure which equation to use. I understand the principal behind buoyancy, but I'm afraid I'm stuck implementing that understanding mathematically.

Thanks.

 

[BruceW]

You're on the right track. You need to write down the equations for buoyant force and weight specifically, and see what they tell you.

I know that since the boat floats with half of it submerged in regular water that the volume of half of the boat is equal to the buoyant force.

You're right that only half of the volume is contributing to the buoyant force, but the buoyant force is not equal to the displaced volume.