1. Using the Power Property with Exponential Models to Make Predictions
Section 11.5
Using the Power Property
with Exponential Models
to Make Predictions

2. Using the Power Property with Exponential Models to Make Predictions
Using the Power Property to Make a Prediction
Example
A person invests $7000 in a bank account with a yearly interest rate of 6%, compounded annually. When will the balance be $10,000?
Let B = f (t) be the balance (in thousand s of dollars) after t years or a fraction thereof
Exponential model of the form f (t) = abt
B-intercept of (0, 7): $7000, when t = 0,
a = 7 and f (t) = 7bt
Solution

3. Using the Power Property with Exponential Models to Make Predictions
Using the Power Property to Make a Prediction
Solution Continued
End of the year, account increased by 6% of previous years balance
Thus, f (t) = 7(1.06)t
To find when the balance is $10,000 (B = 10), substitute 10 for f (t) and solve for t

4. Solution Continued Checking solution:
Using the Power Property with Exponential Models to Make Predictions
Using the Power Property to Make a Prediction
Solution Continued
Checking solution:

5. Using the Power Property with Exponential Models to Make Predictions
Using the Power Property to Make a Prediction
Example
The infant mortality rate is the number of deaths of infants under one year old per births. In 1915, the rate was almost 100 deaths per infants, or 1 death per 10 infants. The infant mortality rate has decreased substantially since then (see the table).

6. Solution Scattergraph shows “bent” Exponential model is needed
Using the Power Property with Exponential Models to Make Predictions
Using the Power Property to Make a Prediction
Solution
Scattergraph shows “bent”
Exponential model is needed
Use the points (15, 99.9) and (100, 6.9)

7. Example Example Continued
Using the Power Property with Exponential Models to Make Predictions
Using the Power Property to Make a Prediction
Example
Example Continued
Let I = f (t) be the infant morality rate (number of deaths per 1000 infants) at t years since Find an equation of f.
What is the percentage rate of decay for infant morality rates?
Find f -1 (5). What does the result mean in this situation?

8. Divide the left and right side by and solve for b:
Using the Power Property with Exponential Models to Make Predictions
Using the Power Property to Make a Prediction
Solution Continued
Divide the left and right side by and solve for b:

9. Using the Power Property with Exponential Models to Make Predictions
Using the Power Property to Make a Prediction
Solution Continued
Substitute the coordinate (100, 6.9) into the equation f (t) = a(0.069)t and solve for a
Graphing calculator shows that the model is a good fit

10. Solution Continued The base b is 0.969 1 – 0.969 = 0.031
Using the Power Property with Exponential Models to Make Predictions
Using the Power Property to Make a Prediction
Solution Continued
The base b is 0.969
1 – = 0.031
The model estimates that the infant mortality rate has decayed 3.1% per year
3. f sends values of t to values of I
f -1 sends values of I to values of t

11. Using the Power Property with Exponential Models to Make Predictions
Using the Power Property to Make a Prediction
Solution Continued
f -1 (5) represents the year (since 1900) when the infant mortality rate will be 5 deaths per infants
Find the year by substituting 5 for f (t)

12. Using the Power Property with Exponential Models to Make Predictions
Using the Power Property to Make a Prediction
Solution Continued
Model predicts infant mortality rate will be 5 deaths per 100 infants in 2010.

13. Using the Power Property with Exponential Models to Make Predictions
Using the Power Property to Make a Prediction
Example
In Exercise 34 of Section 4.5, you may have found that f (t) = 0.10(1.37)t, where f (t) is the chip speed (in MHz) at t years since 1971 (see the table) A rule of thumb for estimating
how quickly technological products improve is that they double in speed every 2 years. Use f to estimate doubling time.

14. Solution In 1971 chip speed was 0.10 MHz
Using the Power Property with Exponential Models to Make Predictions
Using the Power Property to Make a Prediction
Solution
In 1971 chip speed was 0.10 MHz
Find the year when the speed was 2(0.10) = MHz

15. Solution Solution Continued
Using the Power Property with Exponential Models to Make Predictions
Using the Power Property to Make a Prediction
Solution
Solution Continued
According to the function, it took 2.20 years to double the 1971 speed.

16. Using the Power Property with Exponential Models to Make Predictions
Using the Power Property to Make a Estimate
Example
A violent volcanic eruption and subsequent collapse of the former Mount Mazama created Crater Lake, the deepest lake in the United States. Scientists found a charcoal sample from a tree that burned in the eruption. If only 39.40% of the carbon-14 remains in the sample, when did Crater Lake form?

17. Using the Power Property with Exponential Models to Make Predictions
Using the Power Property to Make a Estimate
Solution
Let P = f (t) be percentage of carbon-14 remaining at t years after the sample formed
Percentage is halved every 5730 years
Need to find an exponential equation of the form
t =0, 100% (all) of the carbon-14 remained
P-intercept is (0, 100)
a =100 and f (t)=100bt

18. Solution Continued At time t =5730, ½(100)=50% carbon-14 remained
Using the Power Property with Exponential Models to Make Predictions
Using the Power Property to Make a Estimate
Solution Continued
At time t =5730, ½(100)=50% carbon-14 remained
The equation is f (t) = 100( )t.
Estimate the age of the sample
Substitute for f (t) and solve for t

19. The age of Crater Lake (and the sample) is approximately 7697 years.
Using the Power Property with Exponential Models to Make Predictions
Using the Power Property to Make a Estimate
Solution
Solution Continued
The age of Crater Lake (and the sample) is approximately 7697 years.