13 Section 3.1: Exponential Growth and Decay
Example 1
Determine if each of the following functions are exponential. If they are, write them in the form [latex]f(t) = ab^t[/latex].
- [latex]f(t) = 3^{t+1}[/latex]
- [latex]p(t) = 2^t+ 6^t[/latex]
- [latex]g(t) = \frac{2^{3t}}{7}[/latex]
- [latex]k(t) = 27^{\frac{t}{3}}[/latex]
- [latex]h(t) = 4^t \cdot 2^{2t}[/latex]
Show Solution
a. Yes, exponential. [latex]f(t) = 3^{t+1} = 3^t \cdot 3 = 3\cdot 3^t[/latex]
b. Not exponential.
c. Yes, exponential. [latex]g(t) = \frac{2^{3t}}{7} = \frac{(2^{3)^t}}{7} = \frac{1}{7} \cdot 8^t[/latex].
d. Yes, exponential. [latex]k(t) = 27^{\frac{t}{3}} = (27^{1/3})^t = 3^t.[/latex]
e. Yes, exponential. [latex]h(t) = 4^t \cdot 2^{2t} = 4^t \cdot (2^2)^t = 4^t \cdot 4^t = 4^{t+t} = 4^{2t} = (4^2)^t = 16^t.[/latex]
Example 2
Supposes we have two cities growing at different rates. The population in millions at time t in years for city A is modeled by [latex]P(t) = 120(1.036)^t[/latex] and the population in millions at time t in years for city B is modeled by [latex]Q(t) = 99(1.039)^t[/latex].
a. Which city is growing at a faster rate? What is the annual growth rate of that town?
b. Which city has the largest initial population?
c. After 20 years which town will have more people?
Show Solution
a. City B is growing at a faster rate. The annual growth rate is 3.9%.
b. City A has the largest initial population of 120 million.
c. Plugging in t = 20 into both models we get [latex]P(20) = 120(1.036)^{20} \approx 243.43[/latex] and [latex]Q(20) = 99(1.039)^{20} \approx 212.79[/latex]. So, city A will have more people after 20 years.
Example 3
Find a formula for an exponential function that contains the points [latex](2,48)[/latex] and [latex](4,768)[/latex].
Show Solution
First plug in both points into the equation [latex]f(t) = ab^t[/latex] to get [latex]768 = ab^4[/latex] and [latex]48 = ab^2[/latex]. Then, take the ratio of the two equations to get [latex]\frac{768}{48} = \frac{ab^4}{ab^2}[/latex]. Then simplifying we get [latex]16 = b^2[/latex], so [latex]b = 4[/latex]. Then plugging [latex]b = 4[/latex] back into equation 2 we get [latex]48 = a(4)^2[/latex]. Solving for a we get [latex]a = 3[/latex]. So, [latex]f(t) = 3 \cdot 4^t[/latex].
Example 4
Suppose a bank account is opened initially with $[latex]3500[/latex] and earns [latex]4.2%[/latex] annually each year. Write a model [latex]P(t)[/latex] that models the money in the bank account after t years.
Show Solution
[latex]P(t) = 3500(1.042)^t[/latex].