Chapter 6 | Applications of Integration
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6.8 EXERCISES True or False ? If true, prove it. If false, find the true answer. 348. The doubling time for y = e ct is ⎛ ⎝ ln(2) ⎞ ⎠ / ⎛ ⎝ ln( c ) ⎞ ⎠ . 349. If you invest $500, an annual rate of interest of 3% yields more money in the first year than a 2.5% continuous rate of interest. 350. If you leave a 100°C pot of tea at room temperature (25°C) and an identical pot in the refrigerator (5°C), with k =0.02, the tea in the refrigerator reaches a drinkable temperature (70°C) more than 5 minutes before the tea at room temperature. 351. If given a half-life of t years, the constant k for y = e kt is calculated by k = ln(1/2)/ t . For the following exercises, use y = y 0 e kt . 352. If a culture of bacteria doubles in 3 hours, how many hours does it take to multiply by 10? 353. If bacteria increase by a factor of 10 in 10 hours, how many hours does it take to increase by 100? 354. How old is a skull that contains one-fifth as much radiocarbon as a modern skull? Note that the half-life of radiocarbon is 5730 years. 355. If a relic contains 90% as much radiocarbon as new material, can it have come from the time of Christ (approximately 2000 years ago)? Note that the half-life of radiocarbon is 5730 years. 356. The population of Cairo grew from 5 million to 10 million in 20 years. Use an exponential model to find when the population was 8 million. 357. The populations of New York and Los Angeles are growing at 1% and 1.4% a year, respectively. Starting from 8 million (New York) and 6 million (Los Angeles), when are the populations equal? 358. Suppose the value of $1 in Japanese yen decreases at 2% per year. Starting from $1= ¥250, when will $1= ¥1? 359. The effect of advertising decays exponentially. If 40% of the population remembers a new product after 3 days, how long will 20% remember it?
360. If y =1000 at t =3 and y =3000 at t =4, what was y 0 at t =0? 361. If y =100 at t =4 and y =10 at t =8, when does y =1? 362. If a bank offers annual interest of 7.5% or continuous interest of 7.25%, which has a better annual yield? 363. What continuous interest rate has the same yield as an annual rate of 9%? 364. If you deposit $5000 at 8% annual interest, how many years can you withdraw $500 (starting after the first year) without running out of money? 365. You are trying to save $50,000 in 20 years for college tuition for your child. If interest is a continuous 10%, how much do you need to invest initially? 366. You are cooling a turkey that was taken out of the oven with an internal temperature of 165°F. After 10 minutes of resting the turkey in a 70°F apartment, the temperature has reached 155°F. What is the temperature of the turkey 20 minutes after taking it out of the oven? 367. You are trying to thaw some vegetables that are at a temperature of 1°F. To thaw vegetables safely, you must put them in the refrigerator, which has an ambient temperature of 44°F. You check on your vegetables 2 hours after putting them in the refrigerator to find that they arenow 12°F. Plot the resulting temperature curve and use it to determine when the vegetables reach 33°F. 368. You are an archaeologist and are given a bone that is claimed to be from a Tyrannosaurus Rex. You know these dinosaurs lived during the Cretaceous Era (146 million years to 65 million years ago), and you find by radiocarbon dating that there is 0.000001% the amount of radiocarbon. Is this bone from the Cretaceous? 369. The spent fuel of a nuclear reactor contains plutonium-239, which has a half-life of 24,000 years. If 1 barrel containing 10kg of plutonium-239 is sealed, how many years must pass until only 10 g of plutonium-239 is left? For the next set of exercises, use the following table, which features the world population by decade.
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