# Math Help - first order ODE problem

1. ## first order ODE problem

I don't know what's wrong with me, and I'm ashame of myself, but could you please help me to solve this problem: there is 1 kg of water in the basin and it's(water's) temperature is 20 degrees. an aluminium object with mass of 0,5 kg, thermal capacity of 0,2 and temperature of 75 degrees. after a minute water became 2 degrees warmer. when the difference between water and the object will be less than 1 degree?

I know this cooling equation but can't see how to use it in this situation.

2. Originally Posted by noteiler
I don't know what's wrong with me, and I'm ashame of myself, but could you please help me to solve this problem: there is 1 kg of water in the basin and it's(water's) temperature is 20 degrees. an aluminium object with mass of 0,5 kg, thermal capacity of 0,2 and temperature of 75 degrees. after a minute water became 2 degrees warmer. when the difference between water and the object will be less than 1 degree?

I know this cooling equation but can't see how to use it in this situation.
Let $T_a(t)$ denote the temperature of the aluminium, and $T_w(t)$ that of the water at $t\ge 0$.

Then (assuming Newton's law of cooling applies):

$\dfrac{dT_a}{dt}=-k(T_a(t)-T_w(t))$

and as the heat lost by the aluminium is equal to that gained by the water (assuming the heat lost by the system is negligible):

$0.5 \times 0.2 \times (75 - T_a(t))= T_w(t)-20$

(here I am assuming the thermal capacity of aluminium given is that relative to water, since otherwise the units make no sense)

CB