\v\v*í!?: SUMMARY Activity 5.3. EXERCISES Activity 5.3

Transcription

1 ACTIVITY 5.3 INFLATION Graph the depreciation formula for the car's value, V = 16(0.85)', as a function of the time from the year of purchase. Use the following grid or a graphing calculator. Extend your graph to include 10 years from the date of purchase. 12. How long will it take for the value of the car to decrease to half its original value? Explain how you determined your answer. SUMMARY Activity 5.3 \v\v*í!?: An exponential function is a function in which die independent variable appears as an exponent of the growth factor [for example, c = 8(1.05)'] or a decay factor [for example, V= 16(.85)']. EXERCISES Activity a. Complete the following table, in which/(x) = (\) x and g(x) = 2 X. x f(x) ew J

2 548 CHAPTER 5 MODELING WITH EXPONENTIAL AND LOGARITHMIC FUNCTIONS b. Use the points in part a to sketch a graph of the given functions/and g on the same coordinate axes. Compare the graphs. List the similarities and differences. c. Use a graphing calculator to graph the functions f(x) = (\) x andg(x) = 2 X given in part a in the window Xmin = 2, Xmax = 2, Ymin = 2, and Ymax = 5. Compare to the graphs you obtained in part b. 2. Suppose the inflation rate is 7% per year and remains the same for the next seven years. a. Complete the following table for the cost of a pair of sneakers that costs $45 now. Round to the nearest cent: t,years FROM NOW c(t), COST OF SNEAKERS ($) b. Determine the growth factor for a 7% inflation rate. c. If the yearly inflation rate remains at 7%, what exponential function would you use to determine the cost of $45 sneakers after t years?

3 S'"' ACTIVITY 5.3 INFLATION 549 d. What would be the cost of the sneakers in ten years if the inflation rate stayed at 7%? 3. An exponential function may be increasing or decreasing. Determine which is the case for each of the following functions. Explain how you determined each answer. a. y = 5 X b- y = (Ï)* - 1 c' c. y = 1.5 d. y = Q.2 p 4. a. Evaluate the functions in the following table for the input values, x: g(x) = 3x f(x) = 3* 0 ' ''VVT-V'.-" b. Compare the functions f(x) = 3 X and g(x) = 3x from x = 0 to x = 5 by comparing the output values in the table. m c. Compare the graphs of the functions/and g that are shown in the following window. Approximate the interval in which the exponential function/ grows slower than the linear function g and the interval where it grows faster. WINDOW Xmin=0 Xnax-2 Xscl=.5 Vnin=0 Vn«c*6 Ys-ol=l Xres=l d. Compare the rate of increase of the functions/( x) 3*andg(;t) = 3* from* = 0 to x = 5 by calculating the average rate of change for each function from x = 0 to x = 5. Determine which function grows faster on average in the given interval.

4 550 CHAPTER 5 MODELING WITH EXPONENTIAL AND LOGARITHMIC FUNCTIONS 5. a. For investment purposes, you recently bought a house for $100,000 in a neighborhood where the price of houses has been rising $ 10,000 a year. If the rise in housing prices continues in the same manner, how much will your investment be worth in a year and in two years? b. You decided to buy a second investment house, again for $100,000 but in another area. Here the prices have been rising at a rate of 10% a year. If this rate of increase continues, how much will your investment be worth in a year and in two years? c. Which investment will give you the better return? 6. As a radiology specialist you use the radioactive substance iodine-131 to diagnose conditions of the thyroid gland. Iodine-131 decays (loses radiation energy and changes to a nonradioactive form of iodine) at the rate of 8.3 percent per day. Your hospital currently has a 20-gram supply. a. What is the decay factor for the decay of the iodine? b. Use the decay factor from part a to determine the number of grams remaining for the days listed in the following table. Record your results in the table to the nearest hundredth. t (NUMBER OF DAYS STARTING FROM A 20-GRAM SUPPLY OF IODINE-131) N, NUMBER OF GRAMS OF IODINE-131 REMAINING FROM A 20-GRAM SUPPLY c. Write an exponential decay formula for N, Lhe number of grams of iodine-131 remaining, in terms of i, the number of days from the current supply of 20 grams. d. Determine Lhe number of grams of iodine-131 remaining from a 20-gram supply after 2 months (60 days).

5 ACTIVITY 5.3 INFLATION 551. e. Graph the decay formula for iodine-131, N = 20(0.917j' as a function of the time t (days). Use appropriate scales and labels on the following grid or an appropriate window on a graphing calculator. f. How long will it take for iodine-131 to decrease to half its original value? Explain how you determined your answer. I'