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Unformatted text preview: 67 1.8 GROWTH AND DECAY PROBLEMS 65 contain only 80% of the original amount of radioactive material. room is w. a constant. Find the desired a) What is the half-life of this isotope? room temperature as a function of w and graph it. b) How many additional years will it take until only 15%% of the original amount is left? 23. An object at 100&quot;C is to be placed in a 40'C room. What should the constant of proportion- 20. At the time an item was produced, 0.01 of the ality be in Eq. (25) in order that the object be carbon it contained was carbon-14, a radioiso- at 60&quot;C after 10 min? tope with a half-life of about 5745 yr. 24. The air in a room is cooling. At time / (in hrs) a) You examine the item and discover that the air temperature is Co(t) = 70 + 20e-/ only 0.0001 of the carbon is carbon-14. How An object is placed in the room at time r = 0. old is the object? (This process of determin- The object is initially at 50C and changes ing the age of an object from the amount of temperature according to Eq. (25) with k = 1. carbon-14 it contains is known as carbon-14 a) Find the temperature, T(r), of the object dating.) for 0 s t s 5. b) Derive a formula that gives the age A of b) Graph both Co and T on the same axes. the object in terms of the fraction of carbon that is carbon-14 at the present time, T. 25. An instrument at an initial temperature of 40'C is placed in a room whose temperature 21. The temperature of an engine at the time it is is 20'C. For the next 5 h the room tempera- shut off is 200&quot;C. The surrounding air temper- ture Qo(f) gradually rises and is given by ature is 30&quot;C. After 10 min have elapsed, the Qo(f) = 20 + 10t, t in hours. surface temperature of the engine is 180&quot;C. a) Give the form the cooling law (Eq. 25) a) How long will it take for the surface tem- takes for the instrument. perature of the engine to cool to 40&quot;C? b) From prior experience, you know that your b) For a given temperature 7 between 200&quot;C instrument cools according to Eq. (25) with and 30'C, let (7) be the time it takes to k = 1 if t is measured in hours. If T() is cool the engine from 200'C to T. [For ex- the surface temperature at time t, solve the ample, /(200) = 0 and 1(40) is the answer to equation in part (a) for T(1). part (a).] Find the formula for {(7) in terms c) Graph Qo() and T() on the same axes for of T and graph the function. (The ambient Osis5. temperature is still 30.C.) Many banks compound daily. Continuous com- 22. Earlier experiments have shown that a certain pounding gives answers that are very close to those component cools in air according to the cool- given by daily compounding. In Exercises 26 ing law (Eq. 25) with constant of proportional- through 31, approximate the effect of daily com- ity 0.2. At the end of the first processing stage, pounding by assuming continuous compounding. the temperature of the component is 120&quot;C. 26. You have \$10,000 and intend to invest it for 5 The component remains for 10 min in a large yr in a bank that offers continuous compound- room and then enters the next processing stage. ing. If you want to have \$15,000 in your ac- At that time the surface temperature is sup- count at the end of these 5 yr, what annual posed to be 60&quot;C. interest rate do you have to get? a) What must the room temperature be for 27. You invest \$2000 in an account paying 6% a the desired cooling to take place? year, compounded daily, on the amount in b) Suppose that the entrance and exit temper- excess of \$500. atures are still set at 120'C and 60&quot;C, re- a) Express this as a differential equation by spectively, but the length of the wait in the assuming continuous compounding....
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• Fall '10
• capretta

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