Exponential growth and decay

exponential growth where k > 0 and a is the initial amount.

exponential decay where k > 0 and a is the initial amount.

Example 1

Identify the initial amount and whether the model is growth or decay.

a.) b.)

Solution .

In part a, the initial amount in this growth model is 1500. Note that when x = 0, or . This confirms that 1500 is the initial amount or amount when x = 0. In part b, the initial amount in this decay model is 800. The two curves are graphed below.

Example 2

When does the population in example 1a reach 15000?

Solution .

Solve . First isolate the exponential part by dividing by 1500. = 10. Now take the natural log of both sides. After using the inverse property, we get . This = 1.151292547

Example 3

Initially there were 5000 elk in a region. In 2 years there are 2000. Assume exponential decay and find the equation that describes this population.

Solution .

We've assumed that the equation has the form and we know that (0, 5000) and (2, 2000) are points on the graph of the decay equation. Since a is the initial population, we know that To find k we will use that y = 2000 when x =2. Making these substitutions gives us To solve this for k , we first divide both sides by 5000. This gives Taking the natural log of each side, yields Using the inverse property, we have Hence = .458145365 The population is then approximated by