364 CHAPTER 5 Exponential and Logarithmic Functions 16. Warming Time of a Beer Stein A beer stein has a temperature of ° 28 F. It is placed in a room with a constant temperature of ° 70 F. After 10 minutes, the temperature of the stein has risen to ° 35 F. What will the temperature of the stein be after 30 minutes? How long will it take the stein to reach a temperature of ° 45 F? (See the hint given for Problem 15.) 17. Decomposition of Chlorine in a Pool Under certain water conditions, the free chlorine (hypochlorous acid, HOCl) in a swimming pool decomposes according to the law of uninhibited decay. After shocking his pool, Ben tested the water and found the amount of free chlorine to be 2.5 parts per million (ppm). Twenty-four hours later, Ben tested the water again and found the amount of free chlorine to be 2.2 ppm. What will be the reading after 3 days (that is, 72 hours)? When the chlorine level reaches 1.0 ppm, Ben must shock the pool again. How long can Ben go before he must shock the pool again? 18. Decomposition of Dinitrogen Pentoxide At ° 45 C, dinitrogen pentoxide ( ) N O2 5 decomposes into nitrous dioxide ( ) NO2 and oxygen ( ) O2 according to the law of uninhibited decay. An initial amount of 0.25 M N O2 5 (M is a measure of concentration known as molarity) decomposes to 0.15 M N O2 5 in 17 minutes.What concentration of N O2 5 will remain after 30 minutes? How long will it take until only 0.01 M N O2 5 remains? 19. Decomposition of Sucrose Reacting with water in an acidic solution at ° 35 C, sucrose ( ) C H O 12 22 11 decomposes into glucose ( ) C H O 6 12 6 and fructose ( ) C H O 6 12 6 * according to the law of uninhibited decay. An initial concentration of 0.40 M of sucrose decomposes to 0.36 M sucrose in 30 minutes. What concentration of sucrose will remain after 2 hours? How long will it take until only 0.10 M sucrose remains? 20. Decomposition of Salt in Water Salt (NaCl) decomposes in water into sodium ( )+ Na and chloride ( )− Cl ions according to the law of uninhibited decay. If the initial amount of salt is 25 kilograms and, after 10 hours, 15 kilograms of salt is left, how much salt is left after 1 day? How long does it take until 1 2 kilogram of salt is left? 21. Radioactivity from Chernobyl After the release of radioactive material into the atmosphere from a nuclear power plant at Chernobyl (Ukraine) in 1986, the hay in Austria was contaminated by iodine 131 (half-life 8 days). If it is safe to feed the hay to cows when 10% of the iodine 131 remains, how long did the farmers need to wait to use this hay? 22. Smartphone Users The number of U.S. smartphone users (in millions) t years after 2010 is given by ( ) = + − P t e 298 1 3.439 t 0.3785 (a) What is the growth rate in the number of U.S. smartphone users? (b) Use a graphing utility to graph ( ) P t . (c) In what year does the number of U.S. smartphone users reach 295 million? Source: Statista, 2022. 10. Radioactive Decay The half-life of radioactive potassium is 1.3 billion years. If 10 grams is present now, how much will be present in 100 years? In 1000 years? 11. Estimating the Age of a Tree A piece of charcoal is found to contain 30% of the carbon-14 that it originally had. (a) When did the tree from which the charcoal came die? Use 5730 years as the half-life of carbon-14. (b) Using a graphing utility, graph the relation between the percentage of carbon-14 remaining and time. (c) Using INTERSECT, determine the time that elapses until half of the carbon-14 remains. (d) Verify the answer found in part (a). 12. Estimating the Age of a Fossil A fossilized leaf contains 70% of its normal amount of carbon-14. (a) How old is the fossil? Use 5730 years as the half-life of carbon-14. (b) Using a graphing utility, graph the relation between the percentage of carbon-14 remaining and time. (c) Using INTERSECT, determine the time that elapses until one-fourth of the carbon-14 remains. (d) Verify the answer found in part (a). 13. Cooling Time of a Pizza A pizza baked at ° 450 F is removed from the oven at 5:00 pm and placed in a room that is a constant ° 70 F. After 5 minutes, the pizza is at ° 300 F. (a) At what time can you begin eating the pizza if you want its temperature to be ° 135 F? (b) Using a graphing utility, graph the relation between temperature and time. (c) Using INTERSECT, determine the time that needs to elapse before the pizza is ° 160 F. (d) TRACE the function for large values of time. What do you notice about y, the temperature? 14. Newton’s Law of Cooling A thermometer reading ° 72 F is placed in a refrigerator where the temperature is a constant ° 38 F. (a) If the thermometer reads ° 60 F after 2 minutes, what will it read after 7 minutes? (b) How long will it take before the thermometer reads ° 39 F? (c) Using a graphing utility, graph the relation between temperature and time. (d) Using INTERSECT, determine the time that must elapse before the thermometer reads ° 45 F. (e) TRACE the function for large values of time. What do you notice about y, the temperature? 15. Newton’s Law of Heating A thermometer reading °8 C is brought into a room with a constant temperature of ° 35 C. (a) If the thermometer reads ° 15 C after 3 minutes, what will it read after being in the room for 5 minutes? For 10 minutes? (b) Graph the relation between temperature and time. TRACE to verify that your answers are correct. [Hint: You need to construct a formula similar to equation (4).] *Author’s Note: Surprisingly, the chemical formulas for glucose and fructose are the same: This is not a typo.
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