SOLUTION: Write a exponential function to model the situation. The predict the value of the function after 7 years(to the nearest whole number) A population of 340 animals that decreases at

Algebra ->  Exponential-and-logarithmic-functions -> SOLUTION: Write a exponential function to model the situation. The predict the value of the function after 7 years(to the nearest whole number) A population of 340 animals that decreases at      Log On


   

Question 712908: Write a exponential function to model the situation. The predict the value of the function after 7 years(to the nearest whole number)
A population of 340 animals that decreases at a annual rate of 18%
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
A general equation for exponential growth (or decay) is:
A+=+A%5B0%5D%2Ar%5Et
where
t = number of units of time (years or days or seconds, etc.)
A = the amount after t units of time have elapsed
A%5B0%5D = the initial amount (IOW: the amount when t = 0)
r = rate of change. If r > 1 then the equation is for growth. If 0 < r < 1 then the equation is for decay.

The trickiest part of these is to get the "r" correct. r should be (1 + % change) or (1 + fraction change) where the %/fraction of change is negative for decreases. And, as always, do not use percents as percents in calculations. Use the fraction or decimal equivalent of the percent instead. Since 18% as a decimal is 0.18 and since this is a decrease we should have:
r = (1 + (-0.18)) = 0.82

The rate is an "annual" rate so our "units of time" will be years. Any value for t then, should be in years.

We are given the initial amount, 340, so we know A%5B0%5D. Replacing A%5B0%5D and the r we have found into the general equation we get:
A+=+340%2A%280.82%29%5Et
We now have an equation which models the situation described in the problem. And we can use it to solve the problem. It asks us to find the number of animals after 7 years. So we can use a 7 (since it is in years) for t. (If the problem had asked us to find the number of animals after 7 months then we would have to convert that to years, 7/12, because our "r" is an annual rate.) Replacing the t with 7 we get:
A+=+340%2A%280.82%29%5E7

I will leave it up to you to use your calculator to simplify this. (Just be sure to start with raising 0.82 to the 7th power before multiplying by 340!) The answer you get will not be a whole number. You should off this number to the nearest whole number since fractional parts of an animal do not make sense.