A flat-bottomed boat that is 3 m long and 2 m wide is floating on a lake. The boat sinks by 1 cm when a man steps into it. Find the mass of the man.

Introduction / Context:
This problem combines ideas from mensuration with the principle of flotation. When a man steps into a boat, the boat sinks a little more into the water because it has to displace additional water to balance the extra weight. By measuring the change in water level, we can determine the mass of the man using the relationship between volume, density and weight.

Given Data / Assumptions:

• Length of boat L = 3 m.
• Breadth of boat B = 2 m.
• Additional sinking depth when man steps in = 1 cm = 0.01 m.
• The bottom of the boat is flat, so the displaced volume is L * B * additional depth.
• Density of water is taken as 1000 kg per m^3.
• The system is in equilibrium, so weight of man equals weight of extra water displaced.

Concept / Approach:
Archimedes principle states that the upward buoyant force on an immersed body equals the weight of the fluid displaced. When the man gets into the boat, the extra volume of water displaced corresponds exactly to the weight of the man. The steps are:
Volume of extra water displaced = L * B * extra depth. Mass of displaced water = volume * density. This mass equals the mass of the man because their weights balance.

Step-by-Step Solution:
Step 1: Compute the volume of extra water displaced. Volume = 3 * 2 * 0.01 = 0.06 m^3. Step 2: Use density of water, 1000 kg per m^3. Mass of displaced water = 0.06 * 1000 = 60 kg. Step 3: By Archimedes principle, mass of man = mass of additional water displaced = 60 kg.

Verification / Alternative check:
We can reason qualitatively: 0.1 m^3 of water would have a mass of 100 kg. Our displaced volume of 0.06 m^3 is a little more than half of 0.1 m^3, so the mass should be slightly more than 50 kg. The computed value of 60 kg fits this expectation, giving us confidence in the calculation.

Why Other Options Are Wrong:
12 kg would correspond to a much smaller displaced volume and is unrealistic for an adult human mass.
72 kg and 96 kg are larger than the computed mass and would require the boat to sink more than the observed 1 cm.
48 kg is smaller than 60 kg and could only occur if either the dimensions or the sinking depth were less than given.

Common Pitfalls:
A common mistake is to forget to convert the sinking depth from centimetres to metres, which would lead to a tenfold error in the volume calculation. Another pitfall is to confuse mass with weight and insert an extra factor of gravitational acceleration when it is not needed, because we are working directly in mass units using density. Careful handling of units and straightforward application of Archimedes principle ensures the correct answer.

Final Answer:
The mass of the man is 60 kg.