Newton's Law of Cooling and ambient temperature

[Fancy Moses]

I'm somewhat familiar with the formula:

$$T(t)=T_{0}+(T_{i}-T_{0})e^{kt}$$

However, what if the ambient temperature is not constant? How would one find the temperature of an object with an ambient temperature that ramps from A to B (steady ramp let's say)?

I'm honestly not sure how to solve for a certain time if the temperature is changing...

Any advice would be greatly appreciated!

 

[kuruman]

To begin with, your exponential must be exp(-kt), otherwise it will blow up as time increases. I am sure this was a typo. Now for the big question. Where did the equation that you posted come from? Answer: It is the solution of the differential equation

$$\frac{dT}{dt}=-k(T-T_0)$$

where T₀ is the (constant) ambient temperature. If the ambient temperature is not constant but a function of time f(t), then you replace T₀ with f(t) in the above differential equation and solve it (if you can).