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20 February, 21:03

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.

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  1. 20 February, 22:12
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    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.
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