Understanding Design Flood Return Period

The question requires calculating the return period for a design flood. This is based on the project's expected life and the acceptable risk associated with the flood event.

The key relationship is between the acceptable risk ($R$), the project's service life ($n$), and the annual probability of exceedance ($P$) of the flood. The return period ($T$) is simply the inverse of the annual probability ($T = 1/P$).

Calculating Annual Probability of Exceedance

The probability that a flood event with annual probability $P$ occurs at least once during a project life $n$ is given by $R = 1 - (1 - P)^n$. We need to find $P$ first.

Rearranging the formula to solve for $P$:

1. Start with the risk formula:

   $R = 1 - (1 - P)^n$

2. Isolate the term with $P$:

   $(1 - P)^n = 1 - R$

3. Raise both sides to the power of $1/n$:

   $1 - P = (1 - R)^{1/n}$

4. Solve for $P$:

   $P = 1 - (1 - R)^{1/n}$

Determining the Design Flood Return Period

Use the given values:

• Project Life ($n$): 25 years
• Acceptable Risk ($R$): 5% = 0.05

Substitute these into the formula for $P$:

$P = 1 - (1 - 0.05)^{1/25}$

$P = 1 - (0.95)^{1/25}$

Calculate the value of $(0.95)^{1/25}$:

$(0.95)^{1/25} \approx 0.997959$

Now, find $P$:

$P \approx 1 - 0.997959 \approx 0.002041$

Finally, calculate the return period ($T$):

$T = 1/P \approx 1 / 0.002041 \approx 490.0$ years

Rounding to the nearest integer gives 490 years. This result falls within the range of 476 to 500 years.