Step 1: Understanding the Question:
We must determine the required average speed of a leaking ship so that it reaches safety exactly as it begins to sink.
The ship is currently located 156 km away from the shore.
Water is entering the ship through a leak at a rate of $2 \frac{1}{3}$ metric tons every $6 \frac{1}{2}$ minutes.
Simultaneously, the onboard pumps are ejecting water at a steady rate of 15 metric tons per hour.
The critical threshold for the ship to sink is an accumulation of exactly 68 metric tons of water.
Step 2: Key Formula or Approach:
We need to calculate the net rate at which water accumulates inside the ship per hour.
Net Accumulation Rate = (Rate of water entering per hour) - (Rate of water pumped out per hour).
Once we have this rate, the total survival time of the ship is calculated by dividing the sinking threshold (68 tons) by the net accumulation rate.
Finally, the required sailing speed is found by dividing the distance to the shore by this total survival time.
Step 3: Detailed Explanation:
First, let us calculate the exact rate at which water enters the ship in tons per minute.
The leak admits $2 \frac{1}{3}$ tons, which is $\frac{7}{3}$ tons, in $6 \frac{1}{2}$ minutes, which is $\frac{13}{2}$ minutes.
The rate of water entering per minute is $\frac{\frac{7}{3}}{\frac{13}{2}} = \frac{7}{3} \times \frac{2}{13} = \frac{14}{39}$ tons per minute.
To find the rate per hour, we multiply this minute rate by 60.
Rate of water entering per hour = $\frac{14}{39} \times 60 = \frac{14 \times 20}{13} = \frac{280}{13}$ tons/hr.
The rate at which the pump removes water is given as 15 tons/hr.
To subtract this easily, we can write 15 as a fraction with a denominator of 13.
$15 = \frac{15 \times 13}{13} = \frac{195}{13}$ tons/hr.
Now we calculate the net rate of water accumulation inside the ship per hour.
Net rate = $\frac{280}{13} - \frac{195}{13} = \frac{85}{13}$ tons/hr.
The ship will sink when 68 tons of water have accumulated inside.
We find the total time before the ship sinks by dividing the total capacity by the net rate.
Time to sink = $\frac{68}{\frac{85}{13}} = \frac{68 \times 13}{85}$ hours.
Notice that both 68 and 85 are perfectly divisible by 17.
$68 = 17 \times 4$ and $85 = 17 \times 5$.
So, the Time to sink = $\frac{4 \times 13}{5} = \frac{52}{5}$ hours.
The ship must cover a distance of 156 km within this exact timeframe to survive.
Required average speed = $\frac{\text{Distance}}{\text{Time}} = \frac{156}{\frac{52}{5}}$ km/hr.
We flip the fraction and multiply.
Required speed = $156 \times \frac{5}{52}$ km/hr.
We can see that 156 divided by 52 is exactly 3.
Required speed = $3 \times 5 = 15$ km/hr.
Step 4: Final Answer:
The average rate of sailing should be 15 km/hr.