# Lesson 3

Understanding Rational Inputs

- Let’s look at exponential functions where the input values are not whole numbers.

### Problem 1

Select **all** solutions to \(m \boldcdot m \boldcdot m = 729\).

\(\sqrt{729}\)

\(\frac{729}{3}\)

\(\frac {\sqrt {729}}{3}\)

\(\frac{1}{3} \sqrt{729}\)

\(729^\frac13\)

\(\sqrt[3]{729}\)

### Problem 2

In a pond, the area that is covered by algae doubles each week. When the algae was first spotted, the area it covered was about 12.5 square meters.

- Find the area, in square meters, covered by algae 10 days after it was spotted. Show your reasoning.
- Explain why we can find the area covered by algae 1 day after it was spotted by multiplying 12.5 by \(\sqrt[7]{2}\).

### Problem 3

The function \(m\), defined by \(m(h) = 300\boldcdot\left(\frac34\right)^h\), represents the amount of a medicine, in milligrams, in a patient’s body. \(h\) represents the number of hours after the medicine is administered.

- What does \(m(0.5)\) represent in this situation?
- This graph represents the function \(m\). Use the graph to estimate \(m(0.5)\).
- Suppose the medicine is administered at noon. Use the graph to estimate the amount of medicine in the body at 4:30 p.m. on the same day.

### Problem 4

The area covered by a lake is 11 square kilometers. It is decreasing exponentially at a rate of 2 percent each year and can be modeled by \(A(t)=11 \boldcdot (0.98)^t\).

- By what factor does the area decrease in 10 years?
- By what factor does the area decrease each month?

### Problem 5

The third and fourth numbers in an exponential sequence are 100 and 500. What are the first and second numbers in this sequence?

### Problem 6

The population of a city in thousands is modeled by the function \(f(t) = 250 \boldcdot (1.01)^t\) where \(t\) is the number of years after 1950. Which of the following are predicted by the model? Select **all** that apply.

The population in 1950 was 250.

The population in 1950 was 250,000.

The population grows by 1 percent each year.

The population in 1951 was 275,000.

The population grows exponentially.