Question 1170005: Ship-Cleaning Robots
When a ship arrives in port, an underwater robot cleans the outside of the hull in order to remove sea
creatures, seaweed and dirt. The robot has magnetic wheels that allow it to crawl over the ship’s steel
hull. Making the hull smooth in this way can save about 8% of fuel on the next voyage. It is important
that the cleaning can be completed before the ship leaves port.
(a) (3 marks) The amount of time taken to offload and reload cargo is 22.7 hours. Robot (Type A) cleans
one side of the ship followed by the other side. The cleaning time depends on the amount of dirt on the
hull and is Normally distributed with a mean of 9.3 hours and a standard deviation of 1.6 hours for one
side of the ship. The time for the second side of the ship is also Normally distributed with a mean of 9.3
hours and a standard deviation of 1.6 hours and is correlated with the time taken for the first side with a
correlation coefficient of 0.85. What is the probability that the robot will have finished cleaning the
ship’s hull before the cargo is offloaded and reloaded?
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Let's break down this problem step-by-step.
**(a) Probability of Finishing Cleaning Before Cargo Handling:**
1. **Define Variables:**
* Let X be the cleaning time for the first side of the ship.
* Let Y be the cleaning time for the second side of the ship.
* X ~ N(μx = 9.3, σx = 1.6)
* Y ~ N(μy = 9.3, σy = 1.6)
* Correlation coefficient (ρ) = 0.85
* Cargo handling time = 22.7 hours
2. **Total Cleaning Time (T):**
* T = X + Y
3. **Mean of Total Cleaning Time (μt):**
* μt = μx + μy = 9.3 + 9.3 = 18.6 hours
4. **Variance of Total Cleaning Time (σt²):**
* σt² = σx² + σy² + 2ρσxσy
* σt² = (1.6)² + (1.6)² + 2(0.85)(1.6)(1.6)
* σt² = 2.56 + 2.56 + 4.352
* σt² = 9.472
5. **Standard Deviation of Total Cleaning Time (σt):**
* σt = √9.472 ≈ 3.078 hours
6. **Distribution of Total Cleaning Time:**
* T ~ N(μt = 18.6, σt = 3.078)
7. **Calculate the Z-score:**
* We want to find P(T < 22.7).
* z = (22.7 - 18.6) / 3.078
* z = 4.1 / 3.078 ≈ 1.332
8. **Find the Probability:**
* P(Z < 1.332) using the standard normal distribution table or a calculator.
* P(Z < 1.332) ≈ 0.9087
9. **Conclusion:**
* The probability that the robot will have finished cleaning the ship's hull before the cargo is offloaded and reloaded is approximately 0.9087 or 90.87%.
**Answer:**
The probability is approximately 0.9087.
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