Certainly! Here’s a step-by-step example of solving a problem involving exponential functions.

Problem:

The population of a certain city grows exponentially. If the initial population is 50,000 and it grows at a rate of 3% per year, what will the population be after 10 years?

Solution:

We can use the formula for exponential growth:

$P(t) = P_0 \cdot e^{rt}$

where:

• $P(t)$ is the population after $t$ years,
• $P_0$ is the initial population,
• $r$ is the growth rate (as a decimal), and
• $t$ is the time in years.

1. Identify the values:

   • $P_0 = 50,000$
   • $r = 0.03$ (3% per year, converted to decimal form)
   • $t = 10$ years

2. Plug these values into the formula:

   $P(10) = 50,000 \cdot e^{(0.03 \cdot 10)}$

3. Simplify the exponent:

   $P(10) = 50,000 \cdot e^{0.3}$

4. Calculate $e^{0.3}$:

   Using a calculator, $e^{0.3} \approx 1.3499$.

5. Complete the calculation:

   $P(10) \approx 50,000 \cdot 1.3499 = 67,495$

6. Answer: The population after 10 years is approximately 67,495.

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Would you like a breakdown of each step, or do you have any questions?

Related Questions

1. What happens to the population if the growth rate changes to 5%?
2. How would you calculate the time needed for the population to double?
3. How does the formula change if we’re given a decay rate instead?
4. Can exponential growth formulas be used for calculating compound interest?
5. How would you set up a model for half-life decay?

Tip:

When using exponential functions, always convert the percentage rate to a decimal by dividing by 100 to use it in the formula.