# 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.

1. 30% increase
2. 30% decrease
3. 2% increase
4. 2% decrease
5. 0.04% increase
6. 0.04% decrease
7. 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.

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

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. A: $$156,\!000\boldcdot\left(0.12\right)^t$$ B: $$156,\!000\boldcdot\left(1.12\right)^t$$ C: $$156,\!000\boldcdot\left(1+0.12\right)^t$$ D: $$156,\!000\boldcdot\left(1-0.12\right)^t$$ E: $$156,\!000 \boldcdot \left(1 + \frac{0.12}{12}\right)^t$$ ### 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?

A:

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

B:

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

C:

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

D:

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

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. 1. Explain how the expression would change if the bank had compounded the interest quarterly for the three years. 2. 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. 1. How much money was in the account when it was opened? 2. Sketch a graph of $$f$$. 3. When does the account value reach$2,000?
(From Unit 5, Lesson 9.)

### Problem 7

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

1. Give an example of a domain for which the average rate of change is a good measure of how the function changes.
2. Give an example of a domain for which the average rate of change is not a good measure of how the function changes.
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, $$n$$.
1. Explain why the equation $$P = 3n + 2$$ defines this function.