Question

A boat having a length 4m and breadth 3m is floating on a lake. The boat sinks by 1cm when a man gets on it. What is the mass of the man?
A. 12kg
B. 120kg
C. 72kg
D. 96kg

Answer 1

Hint: Given is length and breadth of boat. Take depth = 1cm, which gives the volume of water displaced. We know volume = mass/density. Take the density of water as 1000\[kg/{{m}^{3}}\]. Substitute volume and density in the formula to get the mass of the man.

Complete step by step answer:
Given the length of the boat = 4m.
Breadth of the boat = 3m.
It is said that the boat sinks by 1cm when a man gets on it.
Here we need to find the mass or weight of the man who gets on the boat.
\[\therefore \]Depth at which the boat sinks = 1cm = \[{{10}^{-2}}\]m.
Convert cm to meters.
1 meter = 100cm
 \[\therefore \]\[1cm=\dfrac{1}{100}\]meter
\[=\dfrac{1}{{{10}^{2}}}={{10}^{-2}}\]or 0.01m.
We know that volume\[=\dfrac{mass}{density}\].
\[\therefore \]Volume of water displaced = mass of man/ density of water.
The density of water is 1000\[kg/{{m}^{3}}\].
Volume of water displaced = length \[\times \]breadth \[\times \]depth
\[\begin{align}
  & =4m\times 3m\times {{10}^{-2}}m \\
 & =12\times {{10}^{-2}}{{m}^{3}} \\
\end{align}\]
\[\therefore \]Mass of man = Volume of water displaced \[\times \] density of water.
\[=12\times {{10}^{-2}}{{m}^{3}}\times {{10}^{3}}kg/{{m}^{3}}\].
\[\because \]\[{{m}^{3}}\]can be cancelled out from the numerator and denominator.
\[\begin{align}
  & =12\times {{10}^{-2}}\times {{10}^{3}}kg \\
 & =12\times 10kg \\
 & =120kg \\
\end{align}\]
\[\therefore \]Mass of man = 120kg.
Therefore, the boat sinks by 1cm when a man of 120kg gets on it.

So, the correct answer is “Option B”.

Note: The length, breadth and depth should have the same units. In the question length and breadth were in meters and depth in cm. Case should be taken to convert them into the same unit.
You can also convert length and breadth into cm. But it may complicate the answer as the density of water is given is \[kg/{{m}^{3}}\]. So it is wise to convert all into meters.