MHB Solve 1st-Order ODE: Chicken in 375° Oven

[find_the_fun]

I'm going to need a little help with this one. I get an answer but it doesn't make sense. The question states

According to Newton's law of cooling, the time rate of change of temperature T(t) of a body immersed in a medium of constant temperature A is proportional to the difference A-T. That is [math]\frac{dT}{dt}=k(A-T)[/math] where k is a positive constant. A 3lb chicken is initially 50 degrees, is put into a 375 deg oven. After 75 minutes it is found that the chicken is 125 deg. When will it be 150 deg?

I may have gone astray right away. Are we trying to solve the differential equation and find T(t)? I found it to be [math]T=A+\frac{C}{e^{kt}}[/math] and then using initial condition T(0)=50 found [math]T=A+\frac{50-A}{e^{kt}}[/math] Then we know T(75)=125. This let's us solve for k. Hold on a sec, basically I just used one IC to solve for C and another to solve for K. I'm not sure that's right.

 

[MarkFL]

We really don't need to find $T$, what I would do to answer this question is separate variables, use the initial conditions and solve for $kt$, then use the other known point on the curve to solve for $k$, and then you will have $t$ as a function of $T$.