511 4.6 Applications and Models of Exponential Growth and Decay (a) Use the model y = y0e kt to find an exponential function that gives the projected enrollment y in school year t. Let the school year 2013–14 correspond to t = 0, 2014–15 correspond to t = 1, and so on, and use the two points indicated on the graph. (b) Estimate the school year for which the projected enrollment in the district will be 21,500 students. 44. YouTube Views The number of views of a YouTube video increases after the number of hours posted as shown in the table. Number of Hours Number of Views 20 100 25 517 30 2015 35 10,248 (a) Use the model y = y0ekt to find an exponential function that gives projected number of views y after number of hours t. Let hour 20 correspond to t = 0, hour 25 correspond to t = 5, and so on, and use the first and last data values given in the table. (b) Estimate the number of views after 50 hr. Life Sciences (Exercises 45–50) 45. Spread of Disease During an epidemic, the number of people who have never had the disease and who are not immune (they are susceptible) decreases exponentially according to the function ƒ1t2 = 15,000e-0.05t, where t is time in days. Find the number of susceptible people at each time. (a) at the beginning of the epidemic (b) after 10 days (c) after 3 weeks 46. Spread of Disease Refer to Exercise 45. Determine how long it will take, to the nearest day, for the initial number of susceptible people to decrease to half its amount. 47. Growth of Bacteria The growth of bacteria makes it necessary to time-date some food products so that they will be sold and consumed before the bacteria count is too high. Suppose for a certain product the number of bacteria present is given by ƒ1t2 = 500e0.1t, where t is time in days and the value of ƒ1t2 is in millions. Find the number of bacteria, in millions, present at each time. (a) 2 days (b) 4 days (c) 1 week 48. Growth of Bacteria How long will it take the bacteria population in Exercise 47 to double? Round the answer to the nearest tenth. 49. Medication Effectiveness Drug effectiveness decreases over time. If, each hour, a drug is only 90% as effective as the previous hour, at some point the patient will not be receiving enough medication and must receive another dose. If the initial dose was 200 mg and the drug was administered 3 hr ago, the expression 20010.9023, which equals 145.8, represents the amount of effective medication still in the system. (The exponent is equal to the number of hours since the drug was administered.) The amount of medication still available in the system is given by the function ƒ1t2 = 20010.902t. In this model, t is in hours and ƒ1t2 is in milligrams. How long will it take for this initial dose to reach the dangerously low level of 50 mg? Round the answer to the nearest tenth.
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