A ship, 40 kilometres from the shore, springs a leak which admits $$3\frac{3}{4}$$ tonnes of water in 12 minutes. 60 tonnes would suffice to sink her, but the ship's pumps can throw out 12 tonnes of water in one hour. Find the average rate of sailing, so that she may reach the shore just as she begging to sink ?

Solution(By Examveda Team)

Quantity of water let in by the leak in 1 minute :
$$\eqalign{ & = \left( {\frac{{3\frac{3}{4}}}{{12}}} \right){\text{tonnes}} \cr & = \left( {\frac{{15}}{4} \times \frac{1}{{12}}} \right){\text{tonnes}} \cr & = \frac{{15}}{{18}}{\text{tonnes}} \cr} $$
Quantity of water thrown out by the pumps in 1 minute :
$$\eqalign{ & = \left( {\frac{{12}}{{60}}} \right){\text{tonnes}} \cr & = \frac{1}{5}{\text{ tonnes}} \cr} $$
Net quantity of water filled in the ship in 1 min :
$$\eqalign{ & = \left( {\frac{{15}}{{48}} - \frac{1}{5}} \right){\text{tonnes}} \cr & = \frac{{27}}{{240}}{\text{ tonnes}} \cr} $$
$$\frac{{27}}{{240}}$$ tonnes water is filled in 1 minute
60 tonnes water is filled in :
$$\eqalign{ & = \left( {\frac{{240}}{{27}} \times 60} \right){\text{min}} \cr & {\text{ = }}\frac{{1600}}{3}{\text{min}} \cr & = \frac{{80}}{9}{\text{hrs}} \cr} $$
Hence, required speed :
$$\eqalign{ & = \left( {\frac{{40}}{{\frac{{80}}{9}}}} \right){\text{km/hr}} \cr & = \left( {40 \times \frac{9}{{80}}} \right){\text{km/hr}} \cr & = \frac{9}{2}{\text{km/hr}} \cr & = 4\frac{1}{2}{\text{km/hr}} \cr} $$