9 13 pts A cup of hot coffee was placed in a room maintained

9. (13 pts) A cup of hot coffee was placed in a room maintained at a constant temperature of 69 degrees, and the coffee temperature was recorded periodically Using exponential regression with the data set, the following temperature model was produced ts Time | C= Coffee | ElapsedTemperature minutes) (degrees F) 166.0 C-69+89.976 e-0,023 10 1252 110.3 where 1 = Time Elapsed (minutes) and C(t) = Coffee Temperature (in degrees) at time t. 201 30 40 50 98.4 60 93.9 (a) Use the temperature model to estimate the coffee temperature C when 13 minutes have elapsed. Report your estimated temperature to the nearest tenth of a degree. Show some work. (b) Suppose the coffee temperature C is 100 degrees. Solve an appropriate equation for to determine how much time has elapsed. Show algebraic work to solve the equation. Report your answer to the nearest tenth of a minute (c) Briefly describe what happens to the coffee temperature C() as the elapsed time t becomes larger and larger. Is this a realistic description of the behavior of the actual coffee temperature?

Solution

C(t) = 69 + 89.976*e^(-0.023*t)

a)   At t = 13 min

C(t) = 69 + 89.976*e^(-0.023*13) = 135.72 degree

b) 100 = 69 + 89.976*e^(-0.023*t)

   t = 46.3 min

c) As the time t becomes larger and larger, C(t) reaches to room temperature (69 degree)