Final answer:
The risk of design failure of a concrete box culvert designed for a 50-year storm over a 75-year service life is approximately 0.278, indicating a 27.8% chance of experiencing failure at least once during its service life.
Step-by-step explanation:
To calculate the risk of design failure of the box culvert for a 50-year storm over its 75-year service life, we use probability concepts. The risk of failure in any given year is 1/50, since the culvert is designed to handle up to a 50-year storm. To find the risk of at least one failure over the 75-year lifespan, we calculate the probability of the culvert surviving each year and then find the compliment.
The probability of survival each year is 1 - (1/50), or 49/50. Over 75 years, the probability of surviving every year without failure is (49/50)^75. To calculate the risk of at least one failure over the 75 years, we subtract this survival probability from 1:
- Calculate annual survival probability: (49/50).
- Calculate 75-year survival probability: (49/50)^75.
- Calculate risk of failure: 1 - (49/50)^75.
Substituting the values:
(49/50)^75 ≈ 0.722.
1 - 0.722 ≈ 0.278.
Therefore, the calculated risk of design failure in decimal format is approximately 0.278. This indicates that there is about a 27.8% chance that the culvert will experience failure due to a storm larger than what it's designed for at least once during its 75-year service life