Suppose that an object with an initial temperature 𝑇0 is placed in an atmosphere of temperature 𝑇𝑎. Newton’s Law of Cooling states that the temperature of the object after 𝑡 minutes is given by the function 𝑇(𝑡) = 𝑇𝑎 + (𝑇0 − 𝑇𝑎 )𝑒 −𝑘𝑡, where 𝑘 is a positive constant that depends on the type of object. a. A restaurant serves a cup of coffee at a temperature of 180°𝐹. If the manager of the restaurant keeps the thermostat set at 67°𝐹, what will be the temperature of the coffee after 30 minutes if no one drinks it? (Assume that 𝑘 = 0.065.) Round to the nearest whole number and include units in your answer. b. At the same restaurant, a bowl of chili is served at 185°𝐹. If a customer cannot eat it at a temperature higher than 106°𝐹, how long will he need to wait? (Assume that 𝑘 = 0.13.) Round to the nearest whole number and include units in your answer.
106 = 67 + ( 185 - 67)e(-..13 * t) subtract 67 from both sides
39 = ( 118)e(-.065 * t) divide both sides by 118
(39/118) = e (-.13 * t) take the Ln of both sides
Ln ( 39/118) =Ln e (-.13 * t) and we can write
Ln ( 39/118) = (-.13 * t) Ln e (Ln e =1....we can ignore this )
Ln ( 39/118) = -.13 * t divide both sides by -.13
Ln (39/118) /-.13 = t ≈ 9 min