A ship is 77 km from the store, springs a leak which admits `2 (1)/(4)` ton of water in every `5 (1)/(2)` min. An outlet tank can throw out 12 tons of water per hour. Find at what speed it should move such that when it begins to sink a rescue ship moves with 6 km/hr excape the passengers of the ship if 69 ton of water is enough to sink ?

Options

A. 7 km/hr
B. 8 km/hr
C. 17/2 km/hr
D. 19/2 km/hr

Answer

To solve the problem step by step, we will analyze the information given and calculate the required speed of the ship.

### Step 1: Calculate the rate of water intake
The ship admits \(2 \frac{1}{4}\) tons of water every \(5 \frac{1}{2}\) minutes.

First, convert \(2 \frac{1}{4}\) to an improper fraction:
\[ 2 \frac{1}{4} = \frac{9}{4} \text{ tons} \]
Next, convert \(5 \frac{1}{2}\) minutes to hours:
\[ 5 \frac{1}{2} = \frac{11}{2} \text{ minutes} = \frac{11}{120} \text{ hours} \]

Now, calculate the rate of water intake per hour:
\[ \text{Rate of water intake} = \frac{\frac{9}{4} \text{ tons}}{\frac{11}{120} \text{ hours}} = \frac{9 \times 120}{4 \times 11} = \frac{1080}{44} = \frac{270}{11} \text{ tons/hour} \]

### Step 2: Calculate the rate of water being expelled
The outlet tank can throw out 12 tons of water per hour.

### Step 3: Calculate the net rate of water accumulation
The net rate of water accumulation in the ship is:
\[ \text{Net rate} = \text{Rate of water intake} - \text{Rate of water expelled} \]
\[ \text{Net rate} = \frac{270}{11} - 12 = \frac{270}{11} - \frac{132}{11} = \frac{138}{11} \text{ tons/hour} \]

### Step 4: Calculate the time taken to accumulate 69 tons of water
To find out how long it takes to accumulate 69 tons of water:
\[ \text{Time} = \frac{\text{Total water}}{\text{Net rate}} = \frac{69}{\frac{138}{11}} = 69 \times \frac{11}{138} = \frac{759}{138} \text{ hours} \approx 5.5 \text{ hours} \]

### Step 5: Calculate the distance the ship can travel in that time
The ship is initially 77 km from the shore. Let \(x\) be the speed of the ship in km/hr. The distance traveled in 5.5 hours is:
\[ \text{Distance} = x \times 5.5 \]

### Step 6: Set up the equation for the distance to the shore
The ship needs to cover the distance to the shore before it sinks:
\[ x \times 5.5 = 77 \]
\[ x = \frac{77}{5.5} = 14 \text{ km/hr} \]

### Step 7: Calculate the speed of the rescue ship
The rescue ship moves at 6 km/hr. The effective speed of the sinking ship must be such that it can reach the shore before it sinks.

### Final Calculation
To find the speed at which the ship should move to ensure it reaches the shore:
\[ \text{Speed of the ship} = 14 \text{ km/hr} \]

### Conclusion
The required speed of the ship should be \(14 \text{ km/hr}\).

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