What is the total energy required to heat the material to 80 degrees?

[Dafe]

Hi,

a question at work popped up and it's been too long since I went to school :p

The total energy [Wh] required to heat the system to temperature T is given by f(T)=1.28T. The effect [W] applied to the system is given by g(T)=378-3.16T. How long does it take to heat the material to say 80 degrees?

Hope someone can help me out, thanks.

 

[Dafe]

Here's my try. I'm using wolframalpha for the differentiation and integration...


$$\frac{dh}{dT}=\frac{d}{dT}\left(\frac{f(T)}{g(T)}\right)$$

$$=\frac{d}{dT}\left(\frac{1.28T}{378-3.16T}\right)$$

$$=\frac{4.04T}{(378-3.16T)^2}+\frac{1.28}{378-3.16T}$$

So now I have the change in time required to heat up the material as a function of temperature. I integrate to get the time required:

$$h(T)=\frac{153.1}{378-3.16T}$$

Meh, at least it's an attempt :p