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68. RULE OF 70 Investors...

68. RULE OF 70 Investors are often interested in knowing how long it takes for a particular investment to double. A simple means for making this determination is the “rule of 70,” which says: The doubling time of an investment with an annual interest rate r% compounded continuously is given by d= 70/r

A. For interest rate r, use the formula B = Pert to find the doubling time for r = 4, 6, 9, 10, and 12. In each case, compare the value with the value obtained from the rule of 70.

B.Some people prefer a “rule of 72” and others use a “rule of 69.” Test these alternative rules as in part (a), and write a paragraph on which rule you would prefer to use.

74.FORENSIC SCIENCE The temperature T (in degrees Celsius) of the body of a murder victim found in a room where the air temperature is 20°C is given by T(t) = 20+17e^-0.07t where t is the number of hours after the victim's death.

A. Graph the body temperature T(t) for t ≥ 0. What is the horizontal asymptote of this graph, and what does it represent?

B.What is the temperature of the victim's body after 10 hours? How long does it take for the body's temperature to reach 25°C?

C.Abel Baker is a clerk in the firm of Dewey, Cheatum, and Howe. He comes to work early one morning and finds the corpse of his boss, Will Cheatum, draped across his desk. He calls the police, and at 8 a.m., they determine that the temperature of the corpse is 33°C. Since the last item entered on the victim's notepad was, “Fire that idiot, Baker,” Abel is considered the prime suspect. Actually, Abel is bright enough to have been reading this text in his spare time. He glances at the thermostat to confirm that the room temperature is 20°C. For what time will he need an alibi to establish his innocence?

85.A population model developed by the U.S. Census Bureau uses the formula P(t) = (202.31) / (1+e^(3.938-0.314t)) to estimate the population of the United States (in millions) for every 10th year from the base year 1790. Thus, for instance, t = 0 corresponds to 1790, t = 1 to 1800, t = 10 to 1890, and so on. The model excludes Alaska and Hawaii.

-Use this formula to compute the population of the contiguous United States for the years 1790, 1800, 1830, 1860, 1880, 1900, 1920, 1940, 1960, 1980, 1990, and 2000.

A. For interest rate r, use the formula B = Pert to find the doubling time for r = 4, 6, 9, 10, and 12. In each case, compare the value with the value obtained from the rule of 70.

B.Some people prefer a “rule of 72” and others use a “rule of 69.” Test these alternative rules as in part (a), and write a paragraph on which rule you would prefer to use.

74.FORENSIC SCIENCE The temperature T (in degrees Celsius) of the body of a murder victim found in a room where the air temperature is 20°C is given by T(t) = 20+17e^-0.07t where t is the number of hours after the victim's death.

A. Graph the body temperature T(t) for t ≥ 0. What is the horizontal asymptote of this graph, and what does it represent?

B.What is the temperature of the victim's body after 10 hours? How long does it take for the body's temperature to reach 25°C?

C.Abel Baker is a clerk in the firm of Dewey, Cheatum, and Howe. He comes to work early one morning and finds the corpse of his boss, Will Cheatum, draped across his desk. He calls the police, and at 8 a.m., they determine that the temperature of the corpse is 33°C. Since the last item entered on the victim's notepad was, “Fire that idiot, Baker,” Abel is considered the prime suspect. Actually, Abel is bright enough to have been reading this text in his spare time. He glances at the thermostat to confirm that the room temperature is 20°C. For what time will he need an alibi to establish his innocence?

85.A population model developed by the U.S. Census Bureau uses the formula P(t) = (202.31) / (1+e^(3.938-0.314t)) to estimate the population of the United States (in millions) for every 10th year from the base year 1790. Thus, for instance, t = 0 corresponds to 1790, t = 1 to 1800, t = 10 to 1890, and so on. The model excludes Alaska and Hawaii.

-Use this formula to compute the population of the contiguous United States for the years 1790, 1800, 1830, 1860, 1880, 1900, 1920, 1940, 1960, 1980, 1990, and 2000.

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