Problem: The population of a city is currently 100,000 and is growing at a rate of 5% per year. How long will it take for the population to reach 200,000?
Solution:
- Identify the exponential model: The population grows exponentially, so we use the formula:
- $$P(t) = P_0(1 + r)^t$$
- where \$P(t)\$ is the population at time \$t\$, \$P_0\$ is the initial population, \$r\$ is the growth rate, and \$t\$ is the time in years.
- Substitute the known values: Here, \$P_0 = 100,000\$, \$r = 0.05\$, and we want \$P(t) = 200,000\$.
- $$200,000 = 100,000(1 + 0.05)^t$$
- Solve for \$t\$ using logarithms:
- Divide both sides by 100,000:
- $$2 = (1.05)^t$$
- Take the logarithm of both sides:
- $$\log(2) = \log((1.05)^t)$$
- Use the property of logarithms \$\log(a^b) = b\log(a)\$:
- $$\log(2) = t\log(1.05)$$
- Solve for \$t\$:
- $$t = \frac{\log(2)}{\log(1.05)}$$
- Calculate the value of \$t\$:
- Using a calculator, \$\log(2) \approx 0.3010\$ and \$\log(1.05) \approx 0.0212\$.
- $$t \approx \frac{0.3010}{0.0212} \approx 14.21$$
- Interpret the result: It will take approximately 14.21 years for the population to reach 200,000.
- A car depreciates in value by 15% each year. If the car is currently worth \$20,000, how long will it take for the car to be worth \$10,000?
- A bacteria culture doubles in size every 3 hours. If the initial population is 500, how long will it take for the population to reach 8,000?
