SECTION 5.8 Exponential Growth and Decay Models; Newton’s Law; Logistic Growth and Decay Models 359 3 Use Newton’s Law of Cooling Newton’s Law of Cooling * states that the temperature of a heated object decreases exponentially over time toward the temperature of the surrounding medium. Newton’s Law of Cooling The temperature u of a heated object at a given time t can be modeled by the following function: ( ) ( ) = + − < u t T u T e k 0 kt 0 (4) where T is the constant temperature of the surrounding medium, u0 is the initial temperature of the heated object, and k is a negative constant. Using Newton’s Law of Cooling An object is heated to ° 100 C (degrees Celsius) and is then allowed to cool in a room whose air temperature is ° 30 C. (a) If the temperature of the object is ° 80 C after 5 minutes, when will its temperature be ° 50 C? (b) Using a graphing utility, graph the relation found between the temperature and time. (c) Using a graphing utility, verify the results from part (a). (d) Using a graphing utility, determine the elapsed time before the object is ° 35 C. (e) What do you notice about the temperature as time passes? Solution EXAMPLE 4 (a) Using formula (4) with = T 30 and = u 100, 0 the temperature (in degrees Celsius) of the object at time t (in minutes) is ( ) ( ) = + − = + u t e e 30 100 30 30 70 kt kt (5) where k is a negative constant. To find k, use the fact that = u 80 when = t 5 [that is, ( ) = u 5 80 ]. Then = + = = = = ≈ − ⋅ e e e k k 80 30 70 50 70 50 70 5 ln 5 7 1 5 ln 5 7 0.0673 k k k 5 5 5 ( ) = u 5 80 Simplify. Solve for e . k5 Rewrite as a logarithm. Solve for k. Formula (5), therefore, becomes ( ) = + − u t e 30 70 . t 0.0673 To find t when = ° u 50 C, solve the equation = + = = − = = − ≈ − − − e e e t t 50 30 70 20 70 20 70 0.0673 ln 2 7 ln 2 7 0.0673 18.6 minutes t t t 0.0673 0.0673 0.0673 Subtract 30 from both sides. Solve for −e t 0.0673 Rewrite as a logarithm. Solve for t. The temperature of the object will be ° 50 C after about 18.6 minutes, or 18 minutes and 37 seconds. *Named after Sir Isaac Newton ( ) 1643–1727 , one of the cofounders of calculus. (continued)
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