Final answer:
To solve the separable differential equation, we separate the variables and integrate both sides, apply initial conditions to solve for the integration constant and rate constant, and obtain the function T(t)=74+(251)e^(kt), which describes the temperature of the cookies as a function of time.
Step-by-step explanation:
The differential equation given is T'=k(T−74), with initial conditions T(0)=325 and T(7)=165. To solve this separable differential equation, we need to separate the variables and integrate both sides. The separation of variables gives us dT/(T-74) = k dt. By integrating both sides, we obtain ∫ dT/(T-74) = ∫ k dt which leads to ln|T-74| = kt + C, where C is the integration constant.
To solve for C, use T(0)=325: ln|325-74| = k(0) + C, thus C = ln|251|. To find k, use T(7)=165: ln|165-74| = k(7) + ln|251|, yielding k after calculations. The final result after solving for k and substituting it back is T(t) = 74 + (251)e^(kt), where e is the base of natural logarithms and k is the constant found from the conditions provided.
The exact value of k is found from this equation by plugging in the values for t at 7 minutes and the corresponding temperature, then solving for k algebraically. Once we have the value of k, we substitute it back into the equation for T(t) to get our final temperature function with respect to time.