# How do you determine how much carbon should be left when you have the time, initial amount, and the decay rate?initial amount= 100 time= 1977 Decay rate= 0.000121

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### 1 Answer

Given that the inital amount of carbon is 100 units, the time is 1977 years, and the decay rate is .000121 units/year, determine the amount of carbon remaining.

This is an exponential decay function of the form `A=A_0(1-r)^t` where `A_0` is the initial amount, `r` is the decay rate, and `t` is the time measured in appropriate units. In this case `A_0=100` , `r=.000121` units per year, and `t` is 1977 years. (If the problem was trying to find the amount of carbon left from a sample weighing 100 units in 1977, then t=2012-1977=35)

**Then `A=100(1-.000121)^1977~~78.732` units.**

**An alternative approach is to use the exponential function `e^x` which is an exponential growth function if x>0, and exponential decay if x<0. Then the function is `A=e^(rt)` and we would get `A=e^(-.000121*1977)~~78.724` units. Notice we used -.000121 as this is a decay situation.