Using the Power Property with Exponential Models to Make Predictions

1. Section 11.4 Using the Power Property with Exponential Models to Make Predictions

2. 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 Using the Power Property with Exponential Models to Make Predictions Using the Power Property to Make a Prediction

3. 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 Using the Power Property with Exponential Models to Make Predictions Using the Power Property to Make a Prediction

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

5. Example The infant mortality rate is the number of deaths of infants under one year old per 1000 births. In 1915, the rate was almost 100 deaths per 1000 infants, or 1 death per 10 infants. The infant mortality rate has decreased substantially since then (see the table). Using the Power Property with Exponential Models to Make Predictions Using the Power Property to Make a Prediction

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

7. Example Example Continued Let I = f (t) be the infant morality rate (number of deaths per 1000 infants) at t years since 1900. 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? Using the Power Property with Exponential Models to Make Predictions Using the Power Property to Make a Prediction

8. Solution Continued 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

9. 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 Using the Power Property with Exponential Models to Make Predictions Using the Power Property to Make a Prediction

10. Solution Continued The base b is 0.969 1 – 0.969 = 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 Using the Power Property with Exponential Models to Make Predictions Using the Power Property to Make a Prediction

11. Solution Continued f-1 (5) represents the year (since 1900) when the infant mortality rate will be 5 deaths per 1000 infants Find the year by substituting 5 for f (t) Using the Power Property with Exponential Models to Make Predictions Using the Power Property to Make a Prediction

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

13. 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. Using the Power Property with Exponential Models to Make Predictions Using the Power Property to Make a Prediction

14. Solution In 1971 chip speed was 0.10 MHz Find the year when the speed was 2(0.10) = 0.20 MHz Using the Power Property with Exponential Models to Make Predictions Using the Power Property to Make a Prediction

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

16. 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? Using the Power Property with Exponential Models to Make Predictions Using the Power Property to Make a Estimate

17. 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 Using the Power Property with Exponential Models to Make Predictions Using the Power Property to Make a Estimate

18. Solution Solution Continued At time t =5730, ½(100)=50% carbon-14 remained The equation is f (t) = 100(0.0999879)t. Estimate the age of the sample Substitute 39.40 for f (t) and solve for t Using the Power Property with Exponential Models to Make Predictions Using the Power Property to Make a Estimate

19. Solution Solution Continued 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