Question

The boat will sink when the weight on it increases beyond 350 kg. There is a hole in it through which the water leaks in at the rate of 0.4 kg/s. The weight of the boat is 1200 kg, and the weight of the boatman is 48 kg. The boatman throws out water at the rate of 0.04 kg/s. There are four passengers whose weights are 42.5 kg, 53.5 kg, 43.5 kg and 54.5 kg. How long will the boat float?

• A. 60 hours
• B. 80 hours
• C. 96 hours
• D. 100 hours
• E. 120 hours

Answer 1

The Correct Option is D

Step 1: Understand the problem.
We are given that the boat will sink when the weight on it increases beyond 350 kg. The boat has a hole, and water leaks in at the rate of 0.4 kg/s. The weight of the boat is 1200 kg, and the weight of the boatman is 48 kg. The boatman throws out water at the rate of 0.04 kg/s. Additionally, there are four passengers with the following weights: 42.5 kg, 53.5 kg, 43.5 kg, and 54.5 kg. We are asked to find how long the boat will float before it sinks.

Step 2: Calculate the total weight of the boat and the passengers.
The total weight on the boat at the start is the sum of the weight of the boat, the boatman, and the passengers:
- Weight of the boat = 1200 kg
- Weight of the boatman = 48 kg
- Total weight of the passengers = \( 42.5 + 53.5 + 43.5 + 54.5 = 194 \) kg
So, the total initial weight on the boat is: \[ 1200 + 48 + 194 = 1442 \, \text{kg} \]

Step 3: Calculate the effective weight being added to the boat.
The water is leaking into the boat at the rate of 0.4 kg/s, and the boatman is throwing out water at the rate of 0.04 kg/s. Therefore, the effective rate at which weight is increasing on the boat is: \[ 0.4 - 0.04 = 0.36 \, \text{kg/s} \] The boat will sink when the total weight exceeds 350 kg above the initial weight. The current weight on the boat is 1442 kg, and it will sink when the weight exceeds 350 kg over the weight of the boat.

Step 4: Determine the time before the boat sinks.
The boat will float until the weight increases by \( 350 - 1442 = 350 \, \text{kg} \). The weight is increasing at the rate of 0.36 kg/s. Therefore, the time \( t \) taken for the weight to increase by 350 kg is: \[ t = \frac{350}{0.36} = 972.22 \, \text{seconds} \] Converting this time to hours: \[ t = \frac{972.22}{3600} \approx 100 \, \text{hours} \]

Step 5: Conclusion.
The boat will float for approximately 100 hours before it sinks.

Final Answer:
The correct answer is (D): 100 hours.