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27.y2e0.4581t28.y1.1e0.3344t29.Mean Life of Radioactive NucleiPhysicists using theradioactive decay equation yy0ektcall the number 1kthemean lifeof a radioactive nucleus. The mean life of a radon-222nucleus is about 10.185.6 days. The mean life of a carbon-14nucleus is more than 8000 years. Show that 95% of the radioactivenuclei originally present in any sample will disintegrate withinthree mean lifetimes, that is, by time t3k. Thus, the meanlife of a nucleus gives a quick way to estimate how long theradioactivity of a sample will last.30.Finding the Original Temperature of a BeamAnaluminum beam was brought from the outside cold into a machineshop where the temperature was held at 65°F. After 10 min, thebeam warmed to 35°F and after another 10 min its temperaturewas 50°F. Use Newton’s Law of Cooling to estimate the beam’sinitial temperature.5°F31.Cooling SoupSuppose that a cup of soup cooled from 90°Cto 60°C in 10 min in a room whose temperature was 20°C. UseNewton’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°Csoup is put into a freezer whose temperature is 15°C. Howlong will it take the soup to cool from 90°C to 35°C?32.Cooling SilverThe temperature of an ingot of silver is 60°Cabove room temperature right now. Twenty minutes ago, it was70°C above room temperature. How far above room temperaturewill 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 CBL™temperature 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°Cdata.T Ts79.47(0.932t)34.A Very Cool ExperimentA temperature probe is removedfrom a cup of hot chocolate and placed in ice water (temperatureTs0°C). The data in Table 6.3 were collected over the next 30 seconds.(a)Writing to LearnExplain why temperature in this experi-ment can be modeled as an exponential function of time.(b)Use exponential regression to find the best exponentialmodel. 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 probewas removed.