Question

1. if the growth rate is 4%, what is the growth factor?

1.04

1. if the decay rate is 8%, what is the decay factor?

0.92

1. if the growth factor is 1.012, what is the growth rate?

1.2%

1. if the decay factor is 0.87, what is the decay rate?

13%
the table shows the yearly profits of an airline company. t = 0 corresponds to 1997.

t, years since 1997	0	1	2	3	4
profit (millions of dollars)	2	2.2	2.42	2.662	2.9282

1. explain how the data indicate that the companys profits are increasing exponentially.

the ratio remains growing by 1.1 constantly

1. what is the growth factor?

1.1

1. let p(t) represent the yearly profit t years since 1997. write the exponential function rule for p(t) that models the yearly profit.

p(t)=2×(1.1)^t

1. a. the yearly profits of this company are increasing by what percent?

0.11
b. this percent is called the

1. the value of your new car t years from now is given by v(t)=18600(0.85)^t.

a. what is the initial value of your car?
b. what is the decay rate? what is the practical meaning of this decay rate in this situation

Explanation:

Step1: Recall growth - factor and growth - rate relationship

The growth factor $b$ for a growth rate $r$ (as a decimal) is given by $b = 1 + r$. For a decay factor $b$ and decay rate $r$ (as a decimal), $b=1 - r$.

Step2: Solve problem 1

Given growth rate $r = 0.04$, then growth factor $b=1 + 0.04=1.04$.

Step3: Solve problem 2

Given decay rate $r = 0.08$, then decay factor $b = 1-0.08 = 0.92$.

Step4: Solve problem 3

Given growth factor $b = 1.012$, then $1.012=1 + r$. Solving for $r$, we get $r=1.012 - 1=0.012$ or $1.2\%$.

Step5: Solve problem 4

Given decay factor $b = 0.87$, then $0.87 = 1 - r$. Solving for $r$, we get $r=1 - 0.87=0.13$ or $13\%$.

Step6: Solve problem 5

To check for exponential growth, we divide consecutive profit values. $\frac{2.2}{2}=1.1$, $\frac{2.42}{2.2}=1.1$, $\frac{2.662}{2.42}=1.1$, $\frac{2.9282}{2.662}=1.1$. Since the ratio of consecutive terms is constant ($1.1$), the data indicates exponential growth.

Step7: Solve problem 6

As shown in problem 5, the growth factor is $1.1$.

Step8: Solve problem 7

The general form of an exponential function is $P(t)=P_0b^t$, where $P_0$ is the initial - value and $b$ is the growth factor. Here, $P_0 = 2$ (profit in 1997) and $b = 1.1$, so $P(t)=2\times(1.1)^t$.

Step9: Solve problem 8a

The growth factor $b = 1.1$, and since $b = 1 + r$, then $r=b - 1=1.1-1 = 0.1$ or $10\%$.

Step10: Solve problem 8b

This percent is called the growth rate.

Step11: Solve problem 9a

For the function $V(t)=18600(0.85)^t$, when $t = 0$, $V(0)=18600(0.85)^0=18600$. So the initial value of the car is $\$18600$.

Step12: Solve problem 9b

The decay factor is $0.85$. Since $b = 1 - r$, then $0.85=1 - r$, and $r=1 - 0.85=0.15$ or $15\%$. The practical meaning is that the value of the car decreases by $15\%$ each year.

Answer:

1. $1.04$
2. $0.92$
3. $1.2\%$
4. $13\%$
5. The ratio of consecutive profit - values is constant ($1.1$).
6. $1.1$
7. $P(t)=2\times(1.1)^t$
8. a. $10\%$

b. growth rate

1. a. $\$18600$

b. $15\%$; The value of the car decreases by $15\%$ each year.