Exponential growth

Question

The function of F defined by F(x) = 31416 (1.24)^x describes the number of mold spores found growing on the pumpkin pie x days after you discovered the mold.

What is the daily rate of growth of this population? (Does this need to be expressed as a decimal or %?)

What is the hourly rate of growth?

Let G(x) be the spore count on the pie x hours after you discovered the mold. Write the description of the function G.

Answer 1

(1) Each day there are 1.24 times as many spores as the day before, that is, there are 24 percent more spores, so the daily rate of growth is 24 percent. (As a decimal, that'd be 0.24, but a rate of growth is almost always expressed as a percent.)

(2) The obvious answer for the hourly rate of growth would be 24 percent divided by 24 = 1 percent, but that's wrong because this is exponential growth: just as the number of new spores in a day depends on the number of spores at the beginning of the day, the number of spores each hour depends on how many spores at the start of the hour. For that to come out 24 percent more after 24 hours, the rate per hour is the 24th root of 1.24. That's 1.24^(1/24) = about 1.009. (To check, 1.009^24 = about 1.2399.) Each hour there are 0.9 percent more spores, so the hourly rate of growth is 0.9 percent.

(3) I'm not sure why the question asks for the _description_ of the function G, but here's its _definition_: G(x) = 31416 * (1.24^(1/24))^x = approx. 31416 * 1.009^x .