HW 7.4.3: Applications of Geometric Series 1. A doctor prescribes a 240-milligram (mg), pain-reducing drug to a patient who has chronic pain. The medical instructions read that this drug should be taken every 4 hours. After 4 hours, 60% of the original dose leaves the body. Under these conditions, the amount of drug remaining in the body, at 4-hour intervals, forms a geometric series. a. We can track the medicine levels in the human body using a geometric series. What is the common ratio, and what is the a, term? b. How is n, the dose number, related to time "t" (in hours)? c. How many milligrams of the drug are present in the body after 4 hours? This is just after the second dose of medicine. d. Fill in the table below. t (hours) 0 4 8 12 16 mg in body |240 e. Using your data spreadsheet, how many milligrams of medicine is remaining in the patient just after the dose given at 24 hours? 72 hours? f. How many milligrams of the drug are remaining in the body after "t" hours (or "n" doses)? g. The minimum lethal dosage of this pain-reducing drug is 600 mg. If the patient is following instructions and continuously (infinitely) takes the drug as prescribed (4 hours in between doses), will the patient ever have this much of the drug in their body? 2. If a ball is dropped from a 12-foot roof and bounces two thirds as high on each successive bounce, find the total number of feet that the ball will travel, including all the bounces.3. A pendulum is released to swing freely. On the first swing, the pendulum travels a distance of 18 inches. On each successive swing, the pendulum travels 90% of the distance of the previous swing 1 swing. What is the total distance the pendulum swings? After how 18 swing 2 18/0.9 many swings has the pendulum traveled 80% of its total distance? swing 3 1810.9 4. Zeno's Paradox: Can the Greek hero Achilles, running at 20 feet per second, ever catch a tortoise, starting 20 feet away and running at 10 feet per second? The Greek mathematician Zeno said no. He reasoned as follows: When Achilles runs 20 feet the tortoise will be in a new spot, 10 feet away. Then, when Achilles gets to that spot, the tortoise will be 5 feet away. . Achilles will keep cutting the distance in half but will never catch the tortoise. In actuality, looking at the race as Zeno did, you can see that both the distances and the times required to achieve them, form infinite geometric series. Using the table, show that both series have finite sums. What do these sums represent? Distance (ft) 20 10 5 2.5 1.25 0.65 Time (sec) 1 0.5 0.25 0.125 0.0625 0.03125