How To Solve Exponential Decay Problems

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Half-life is the amount of time it takes for a substance to decay to half of the original amount.

Example: A certain isotope has a half-life of 4.2 days.

How long will it take a 150-milligram sample to decay so that only 10 milligrams are left?

Answer: It takes about 1,343.5 years for a bone to lose 15% of its carbon-14. Put together a mathematical model using the initial amount and the exponential rate of growth/decay.

It is best to work from the inside out, starting with the exponent, then the exponential, and finally the multiplication, like this: Not all algebra classes cover this method.

If you're required to use the first method for every exercise of this type, then do so (in order to get the full points). No matter the particular letters used, the green variable stands for the ending amount, the blue variable stands for the beginning amount, the red variable stands for the growth or decay constant, and the purple variable stands for time.Get comfortable with this formula; you'll be seeing a lot of it.The exponential function is the function , exponentiation by e.It is a very important function in analysis, both real and complex.Example: A certain bacterium has an exponential growth rate of 25% per day.If we start with 0.5 gram and provide unlimited resources how much bacteria can we grow in 2 weeks?At this rate how long will it take to grow to 50,000 cells? Example: A certain animal species can double its population every 30 years.Assuming exponential growth, how long will it take the population to grow from 40 specimens to 500? Up to this point, we have seen only exponential growth.Note that the variables may change from one problem to another, or from one context to another, but that the structure of the equation is always the same.For instance, all of the following represent the same relationship: on and so forth.


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