# Exponential Functions - Problem Solving

An **exponential function** is a function of the form $f(x)=a \cdot b^x,$ where $a$ and $b$ are real numbers and $b$ is positive. Exponential functions are used to model relationships with exponential growth or decay. **Exponential growth** occurs when a function's rate of change is proportional to the function's current value. Whenever an exponential function is decreasing, this is often referred to as **exponential decay**.

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## Growth and Decay

Suppose that the population of rabbits increases by 1.5 times a month. When the initial population is 100, what is the approximate integer population after a year?

The population after $n$ months is given by $100 \times 1.5^n.$ Therefore, the approximate population after a year is $100 \times 1.5^{12} \approx 100 \times 129.75 = 12975. \ _\square$

Suppose that the population of rabbits increases by 1.5 times a month. At the end of a month, 10 rabbits immigrate in. When the initial population is 100, what is the approximate integer population after a year?

Let $p(n)$ be the population after $n$ months. Then $p(n+2) = 1.5 p(n+1) + 10$ and $p(n+1) = 1.5 p(n) + 10,$ from which we have $p(n+2) - p(n+1) = 1.5 \big(p(n+1) - p(n)\big).$ Then the population after $n$ months is given by $p(0) + \big(p(1) - p(0)\big) \frac{1.5^{n} - 1}{1.5 - 1} .$ Therefore, the population after a year is given by $\begin{aligned} 100 + (160 - 100) \frac{1.5^{12} - 1}{1.5 - 1} \approx& 100 + 60 \times 257.493 \\ \approx& 15550. \ _\square \end{aligned}$

Suppose that the annual interest is 3 %. When the initial balance is 1,000 dollars, how many years would it take to have 10,000 dollars?

The balance after $n$ years is given by $1000 \times 1.03^n.$ To have the balance 10,000 dollars, we need $\begin{aligned} 1000 \times 1.03^n \ge& 10000 \\ 1.03^n \ge& 10\\ n \log_{10}{1.03} \ge& 1 \\ n \ge& 77.898\dots. \end{aligned}$ Therefore, it would take 78 years. $_\square$

The half-life of carbon-14 is approximately 5730 years. Humans began agriculture approximately ten thousand years ago. If we had 1 kg of carbon-14 at that moment, how much carbon-14 in grams would we have now?

The weight of carbon-14 after $n$ years is given by $1000 \times \left( \frac{1}{2} \right)^{\frac{n}{5730}}$ in grams. Therefore, the weight after 10000 years is given by $1000 \times \left( \frac{1}{2} \right)^{\frac{10000}{5730}} \approx 1000 \times 0.298 = 298.$ Therefore, we would have approximately 298 g. $_\square$

## Problem Solving - Basic

## Problem Solving - Intermediate

## Problem Solving - Advanced

**Cite as:**Exponential Functions - Problem Solving.

*Brilliant.org*. Retrieved from https://brilliant.org/wiki/exponential-functions/