# Applications Involving Exponential Models

Exponential functions are useful in modeling many physical phenomena, such as populations, interest rates, radioactive decay, and the amount of medicine in the bloodstream. An exponential model is of the form A = A_{0}(b)^{t/c} where we have:

- A
_{0}= the initial amount of whatever is being modelled. - t = elapsed time.
- A = the amount at time, t.
- b = the growth factor. Note that if b > 1, then we have exponential growth, and if 0< b < 1, then we have exponential decay.
- c = time it takes for the growth factor b to occur.

**Example:** Suppose that the initial number of bacteria in a sample is 6000 and that the population triples every 2 hours. Set up the corresponding model for the number of bacteria as a function of time.

*Solution:*

f(t) = 6000 ∙ 3^{t/2}

**Example:** If a substance has a half-life of 18 years and we start with 27 mg, determine the appropriate model for how much of the substance we have at any given time.

*Solution:*

f(t) = 27 ∙ (1/2)^{t/18}

Once a model has been set up, known data can be substituted in and the resulting exponential equation can be solved for an unknown variable, which is often t, A, or A_{0}.

**Example 1:**

**Example 2:**

**Example 3:**