A city with a population of 426,000 is forced to evacuate due to a natural disaster. The damage from the disaster caused many residents to move away, and the population decreased exponentially by 26% a year after the disaster. If the population continues to decrease at this rate for t years, write the expression that reveals the monthly rate of decrease for the population in the form (b) c. Round any decimals to the nearest thousandth.

The rate of exponential decay is 26%. What is the rate of decay per month? It is tempting to say the rate per month is. 26 : 12 = 2.1%, but this is not correct. Here is an example using 1 year:

100 * (1-.021) ^12 = 77.4, which implies a rate of decay in one year of 22.6%, not 26%.

We actually must take (1-.26) ^ (1/12), to find the rate of exponential decay per month.

Here's a quick illustration of why this is right:

Initial population: 100

Population after 1 year: 100 * (1-0.26) = 74

Population after 1 month: 100 * (1-.26) ^ (1/12) = 97.52

Population after 12 months: (100) * ((1-.26) ^ (1/12)) ^12 = 74

Therefore:

(1-.26) ^ (1/12) = 0.975

Every month, the population decays by (1-.975) = 2.5%

P = 426,000 * (.975) ^ (12t)

Let's make sure our function is accurate:

When t = 1 year:

P = 426,000 * (.975^12)

P = 426,000 *.74

P = 315, 240

The key here is that. 975^12 is equal to (1-0.26), which confirms that we have found the right monthly rate of decay.