# Lesson 18

Expressed in Different Ways

Let's write exponential expressions in different ways.

### Problem 1

For each growth rate, find the associated growth factor.

- 30% increase
- 30% decrease
- 2% increase
- 2% decrease
- 0.04% increase
- 0.04% decrease
- 100% increase

### Problem 2

In 1990, the population \(p\) of India was about 870.5 million people. By 1995, there were about 960.9 million people. The equation \(p=870.5\boldcdot \left(1.021\right)^t \) approximates the number of people, in millions, in terms of the number of years \(t\) since 1990.

- By what factor does the number of people grow in one year?
- If \(d\) is time in decades, write an equation expressing the number of people, \(p\), in terms of decades, \(d\), since 1990.
- Use the model \(p=870.5\boldcdot\left(1.021\right)^t\) to predict the number of people in India in 2015.
- The 2015, the population of India was 1,311 million. How does this compare with the predicted number?

### Problem 3

An investor paid $156,000 for a condominium in Texas in 2008. The value of the homes in the neighborhood have been appreciating by about 12% annually.

Select **all **the expressions that could be used to calculate the value of the house, in dollars, after \(t\) years.

\(156,\!000\boldcdot\left(0.12\right)^t\)

\(156,\!000\boldcdot\left(1.12\right)^t\)

\(156,\!000\boldcdot\left(1+0.12\right)^t\)

\(156,\!000\boldcdot\left(1-0.12\right)^t\)

\(156,\!000 \boldcdot \left(1 + \frac{0.12}{12}\right)^t\)

### Problem 4

A credit card has a nominal annual interest rate of 18%, and interest is compounded monthly. The cardholder uses the card to make a $30 purchase.

Which expression represents the balance on the card after 5 years, in dollars, assuming no further charges or payments are made?

\(30(1+18)^5\)

\(30(1+0.18)^5\)

\(30\left(1+\frac{0.18}{12}\right)^5\)

\(30\left(1+\frac{0.18}{12}\right)^{5\boldcdot12}\)

### Problem 5

The expression \(1,\!500\cdot\left(1.085\right)^3\) represents an account balance in dollars after three years with an initial deposit of $1,500. The account pays 8.5% interest, compounded annually for three years.

- Explain how the expression would change if the bank had compounded the interest quarterly for the three years.
- Write a new expression to represent the account balance, in dollars, if interest is compounded quarterly.

### Problem 6

The function, \(f\), defined by \(f(t) = 1,\!000 \boldcdot \left(1.07\right)^t\), represents the amount of money in a bank account \(t\) years after it was opened.

- How much money was in the account when it was opened?
- Sketch a graph of \(f\).
- When does the account value reach $2,000?

### Problem 7

The graph shows the number of patients with an infectious disease over a period of 15 weeks.

- Give an example of a domain for which the average rate of change is a good measure of how the function changes.
- Give an example of a domain for which the average rate of change is not a good measure of how the function changes.

### Problem 8

A party will have pentagonal tables placed together. The number of people, \(P\), who can sit at the tables is a function of the number of tables, \(n\).

- Explain why the equation \(P = 3n + 2\) defines this function.
- How many tables are needed if 47 people come to the party?
- How many tables are needed if 99 people come to the party?
- Write the inverse of this function and explain what the inverse function tells us.