# Lesson 5

Changes Over Rational Intervals

### Problem 1

The table shows the monthly revenue of a business rising exponentially since it opened an online store.

months since online
store opened
monthly revenue
in dollars
0 72,000
1
3 90,000
4
6 112,500
1. Describe how the monthly revenue is growing.
2. Write an equation to represent the revenue, $$R$$, as a function of months, $$m$$, since the online store opened.
3. Find the monthly revenue 1 month after the online store opened. Record the value in the table. Explain your reasoning.
4. Explain how we can use the value of $$R(1)$$ to find $$R(4)$$.

### Problem 2

At 7 a.m., a colony of 100 bacteria is placed on a petri dish where the population will triple every 6 hours.

Select all statements that are true about the bacteria population.

A:

When the bacteria population reaches 900, 12 hours have passed since the colony was placed on the petri dish.

B:

Three hours after the colony is placed on the petri dish, there are 200 bacteria.

C:

Three hours after the colony is placed on the petri dish, there are about 173 bacteria in the colony.

D:

In the first hour the colony is placed on the petri dish, the population grows by a factor of $$3^\frac16$$.

E:

Between 8 a.m. and 9 a.m., the population grows by a factor of $$3^\frac23$$.

### Problem 3

The graph represents the cost of a medical treatment, in dollars, as a function of time, $$d$$, in decades since 1978.

Find the cost of the treatment, in dollars, when $$d=1$$. Show your reasoning.

### Problem 4

The exponential function $$f$$ is given by $$f(x) = 3^x$$.

1. By what factor does $$f$$ increase when the exponent $$x$$ increases by 1? Explain how you know.
2. By what factor does $$f$$ increase when the exponent $$x$$ increases by 2? Explain how you know.
3. By what factor does $$f$$ increase when the exponent $$x$$ increases by $$\frac{1}{2}$$? Explain how you know.

### Problem 5

A piece of paper has area 93.5 square inches. How many times does it need to be folded in half before the area is less than 1 square inch? Explain how you know.