Question

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.

Answer 1

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.