A boat having a length of 3 m and a breadth of 2 m is floating on a lake. When a man steps into the boat, it sinks by 1 cm. What is the mass of the man, assuming the density of water is 1000 kg per cubic metre?

Introduction / Context:
This question is a practical application of the principle of flotation. When a man steps into the boat, the boat sinks a little more to displace extra water whose weight is equal to the weight of the man. By computing the extra volume of water displaced, we can find the mass of the man. This type of problem links basic physics with simple volume calculations.

Given Data / Assumptions:

- Length of the boat = 3 m.
- Breadth of the boat = 2 m.
- Boat sinks by 1 cm when the man steps in.
- 1 cm = 0.01 m.
- Density of water is taken as 1000 kg per cubic metre.
- The boat base is assumed to be approximately rectangular and in full contact with the water surface area considered for displacement.

Concept / Approach:
According to the principle of flotation, the weight of water displaced equals the total weight of the floating body and any additional load. Here, the extra volume of water displaced when the man enters the boat multiplied by the density of water gives the mass of the man. Volume displaced is calculated as base area of the boat multiplied by the additional sinking depth.

Step-by-Step Solution:
Step 1: Base area of the boat A = length * breadth = 3 * 2 = 6 sq. m.Step 2: Additional sinking depth h = 1 cm = 0.01 m.Step 3: Extra volume of water displaced V = A * h = 6 * 0.01 = 0.06 cubic metres.Step 4: Mass of water displaced m_water = density * volume = 1000 * 0.06.Step 5: m_water = 60 kg.Step 6: By flotation, the mass of the man must equal the mass of this extra displaced water.Step 7: Therefore, mass of the man = 60 kg.

Verification / Alternative check:
As a sense check, note that a volume of 0.06 cubic metres is 60 litres of water, because 1 cubic metre equals 1000 litres. The mass of 60 litres of water is approximately 60 kg, which is a very reasonable value for a human adult. This supports the correctness of the computed mass.

Why Other Options Are Wrong:
12 kg would correspond to a volume displaced of only 0.012 cubic metres, which does not match the given sinking depth. Values like 48 kg, 72 kg and 90 kg would require different sinking depths or different boat dimensions. They are distractors that result from miscalculating area, forgetting to convert centimetres to metres or using the wrong density.

Common Pitfalls:
Common mistakes include using 1 cm as 0.1 m instead of 0.01 m, or forgetting that density is 1000 kg per cubic metre, not 1 kg per cubic metre. Some learners also mistakenly multiply area and depth in centimetres without converting to metres, which leads to a volume that is off by a factor of 10000. Always keep units consistent and remember that displacement volume times density gives mass.

Final Answer:
Thus, the mass of the man is 60 kg.