Newton’s law of cooling indicates that the temperature of a warm object, such as a cake coming out of the oven, will decrease exponentially with time and will approach the temperature of the surrounding air. The temperature T(t) is modeled by T(t) = Ta + (T0 − Ta)e−kt. In this model, Ta represents the temperature of the surrounding air, T0 represents the initial temperature of the object, and t is the time after the object starts cooling. The value of k is a constant of proportion relating the temperature of the object to its rate of temperature change. Use this model for Exercise. Water in a water heater is originally 122°F. The water heater is shut off and the water cools to the temperature of the surrounding air, which is 60°F. The water cools slowly because of the insulation inside the heater, and the value of k is measured as 0.00351.a. Write a function that models the temperature T(t) (in °F) of the water t hours after the water heater is shut off. b. What is the temperature of the water 12 hr after the heater is shut off? Round to the nearest degree. c. Dominic does not like to shower with water less than 115°F. If Dominic waits 24 hr, will the water still be warm enough for a shower?
Newton’s law of cooling indicates that the temperature of a warm object, such as a cake coming out of the oven, will decrease exponentially with time and will approach the temperature of the surrounding air. The temperature T(t) is modeled by T(t) = Ta + (T0 − Ta)e−kt. In this model, Ta represents the temperature of the surrounding air, T0 represents the initial temperature of the object, and t is the time after the object starts cooling. The value of k is a constant of proportion relating the temperature of the object to its rate of temperature change. Use this model for Exercise. Water in a water heater is originally 122°F. The water heater is shut off and the water cools to the temperature of the surrounding air, which is 60°F. The water cools slowly because of the insulation inside the heater, and the value of k is measured as 0.00351.a. Write a function that models the temperature T(t) (in °F) of the water t hours after the water heater is shut off. b. What is the temperature of the water 12 hr after the heater is shut off? Round to the nearest degree. c. Dominic does not like to shower with water less than 115°F. If Dominic waits 24 hr, will the water still be warm enough for a shower?
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