Calculating the equilibrium temperature of a bob and water system when energy can be lost to the surroundings via heat [closed]

Question

Let's say we have an aluminium bob at X degrees Celsius, and we have a water container at Y degrees Celsius. I am assuming here that the initial temperature of the container and the surroundings is the same, so it's Y.

Now let X>Y

Ideally, we say that the heat lost by the aluminum bob would be the same as the heat gained by the container + Water. But this is when there is no heat lost to the surroundings.

But what if heat was lost to the surroundings?

Let energy lost via heat by aluminum be A; B for energy gained via heat by the container + Water; C for energy gained by the surrounding via heat

So A=B+C, as energy is conserved

Now we can calculate the heat lost to the surroundings by Newton's law of cooling

So, how would we show it mathematically?

If the mass of the bob is M, specific heat capacity is C and equilibrium temperature is T and mass of water is m and its specific heat is c, mass of the container is n, and its specific heat capacity is s

Then, in that case, MC(X-T)=(mc+ns)(T-Y) + k(Y-T)*t [the second term came by integrating the equation of Newton's law wrt to time]

where t is the time it takes for that change, I am assuming we can somehow calculate it

Now I have two questions:

1. Is my reasoning and formulation correct

2. What could be a better way to find the equilibrium in this supposedly non-ideal case