Exponential Growth and Decay – Practice Problems

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Imagine you have a material, like gold or kryptonite, and you have to find out the amount of that material you have after a given percentage of change, after a certain period of time. Instead of doing multiple and repetitive multiplications and additions, there is a shorter and easier way to do it, using the formulas for exponential growth and decay.

The standard formula for exponential growth is X_t = X_0(1 + r)^(t/n), where:
t is the period of time, expressed in units of time,
X_t is the amount after the period of time t,
X_0 is the original size or amount,
r is the rate of growth, also called growth factor,
n is the time needed for 1 growth factor.

Likewise, the standard formula for exponential decay is X_t = X_0(1 - r)^(t/n), where r is now the rate of decay, which is why it is being subtracted now instead of added.

Lets look at an example: a certain culture with 10,000 microorganisms in a petri dish were seemed to be increasing in quantity by 10% after 3 seconds. We can use the formula for exponential growth to find out how many organisms will be in the petri dish after 30 seconds. To do this we first see that
X_0 = 10000,
r = 10% or 0.1,
t = 30 days, and
n = 3 days

So, X_30 = 10000 * (1 + 0.1)^(30/3)
X_30 = 10000 * (1.1)^10
X_30 = 10000 * 2.5937
X_30 = 25,937 microorganisms

Knowing how to compute exponential growth and decay can be useful when you find yourself working in laboratory research, banking and financial analysis, and even agriculture and tech fields that require the use of meaningful statistics. These formulas then become powerful tools, in some sense, for predicting the future.

Analyze functions using different representations.

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Exercises in this Practice Problem
Define exponential growth and decay.
Determine the growth of Bevo's horns.
Calculate the size of Bevo's horns.
Decide if either exponential growth or decay is given.
Establish the formula for Bevo's shrinking horns.
Decide the corresponding exponential function.