27 y 2 e 04581 t 28 y 11 e 03344 t 29 mean life of

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27. y 2 e 0.4581 t 28. y 1.1 e 0.3344 t 29. Mean Life of Radioactive Nuclei Physicists using the radioactive decay equation y y 0 e kt call the number 1 k the mean life of a radioactive nucleus. The mean life of a radon-222 nucleus is about 1 0.18 5.6 days. The mean life of a carbon-14 nucleus is more than 8000 years. Show that 95% of the radioactive nuclei originally present in any sample will disintegrate within three mean lifetimes, that is, by time t 3 k . Thus, the mean life of a nucleus gives a quick way to estimate how long the radioactivity of a sample will last. 30 . Finding the Original Temperature of a Beam An aluminum beam was brought from the outside cold into a machine shop where the temperature was held at 65°F. After 10 min, the beam warmed to 35°F and after another 10 min its temperature was 50°F. Use Newton’s Law of Cooling to estimate the beam’s initial temperature. 5 ° F 31. Cooling Soup Suppose that a cup of soup cooled from 90°C to 60°C in 10 min in a room whose temperature was 20°C. Use Newton’s Law of Cooling to answer the following questions. (a) How much longer would it take the soup to cool to 35°C? (b) Instead of being left to stand in the room, the cup of 90°C soup is put into a freezer whose temperature is 15°C. How long will it take the soup to cool from 90°C to 35°C? 32. Cooling Silver The temperature of an ingot of silver is 60°C above room temperature right now. Twenty minutes ago, it was 70°C above room temperature. How far above room temperature will the silver be (a) 15 minutes from now? 53.45 ° above room temperature (b) 2 hours from now? 23.79 ° above room temperature (c) When will the silver be 10°C above room temperature? 33.Temperature ExperimentA temperature probe is removedfrom a cup of coffee and placed in water whose temperature (Ts)is 10°C. The data in Table 6.2 were collected over the next 30 sec with a CBLtemperature probe. (a)Find an exponential regression equation for the t,TTs(b)Use the regression equation in part (a) to find a model forthe t,Tdata. Superimpose the graph of the model on a scatterplot of the t,Tdata.See answer section.(c)Estimate when the temperature probe will read 12°C.(d)Estimate the coffee’s temperature when the temperatureprobe was removed.89.47°C data. T T s 79.47(0.932 t ) 34. A Very Cool Experiment A temperature probe is removed from a cup of hot chocolate and placed in ice water (temperature T s 0 ° C ). The data in Table 6.3 were collected over the next 30 seconds. (a) Writing to Learn Explain why temperature in this experi- ment can be modeled as an exponential function of time. (b) Use exponential regression to find the best exponential model. Superimpose a graph of the model on a scatter plot of the ( time, temperature ) data. See answer section. (c) Estimate when the probe will reach 5 ° C. At about 37 seconds. (d) Estimate the temperature of the hot chocolate when the probe was removed.

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