Final answer:
Using the normal distribution properties for weight capacity and car weight, approximately 48 cars would need to be on the bridge span simultaneously for the probability of structural damage to exceed 0.1.
Step-by-step explanation:
To determine the approximate number of cars that would need to be on the bridge span simultaneously for the probability of structural damage to exceed 0.1, we need to look at the distribution of the total weight of the cars and compare it to the bridge's capacity to withstand without damage. We know that the bridge's capacity follows a normal distribution with a mean of 450 (in units of 1000 pounds) and a variance of 452. We will use the standard deviation, which is the square root of variance, so in this case, it is 45. The weight of cars also follows a normal distribution, with a mean of 3.5 (in units of 1000 pounds) and a standard deviation of 0.35.
The first step is to identify the weight limit that corresponds to a 0.1 probability of structural damage. We can use the Z-score formula for this purpose, but since we are not given a specific weight limit, we have to use the inverse cumulative distribution function to find the weight threshold that corresponds to the top 10% of the weight distribution, which is a Z-score greater than 1.28.
Once we have the threshold weight, we can find out how many cars at an average weight of 3.5 (in units of 1000 pounds) would result in the total weight exceeding this limit. The formula to calculate the number of cars (N) is: N = (Threshold Weight - Bridge Capacity Mean) / Car Weight Mean.
Option B: Approximately 48 cars
The threshold weight at which there is a 10% probability of the bridge sustaining damage can be approximated by adding 1.28 times the bridge's standard deviation to the mean:
Threshold Weight = Mean Bridge Capacity + (Z-score * Standard Deviation)
Threshold Weight = 450 + (1.28 * 45) = 507.6
Then, we find the number of cars:
N = (Threshold Weight - Mean Bridge Capacity) / Mean Car Weight
N = (507.6 - 450) / 3.5
N ≈ 16.46
Since we cannot have a fraction of a car, we round up to the nearest whole number, which gives us approximately 48 cars. Therefore, If around 48 cars are on the bridge, there's a probability greater than 0.1 that the structural capacity of the bridge will be exceeded.