A hot body dropped into cooler fluid has two resistances in series: conduction out through its own inside, and convection off its surface. The Biot number weighs one against the other: $\mathrm{Bi} = \dfrac{h L_c}{k}$, where $h$ is the film coefficient, $L_c$ the characteristic length in metres (volume divided by surface area), and $k$ is the conductivity of the SOLID — not of the fluid, which is the mistake that ends most attempts.

Below $\mathrm{Bi} = 0.1$, the inside conducts so much better than the surface convects that the body may honestly be treated as sitting at ONE temperature throughout. That is the licence to lump, and it is checked first, every time. Above 0.1 the body carries a real internal gradient and the lumped curve will lie to you — confidently, and with excellent arithmetic.

With the licence in hand, the body gets a thermal time constant: $\tau = \dfrac{\rho V c}{hA}$ — tau equals rho V c over h A. The top is the heat stored per kelvin (density $\rho$ in $\mathrm{kg/m^3}$, volume $V$ in $\mathrm{m^3}$, specific heat $c$), the bottom is the surface conductance in W/K, and $\tau$ comes out in seconds. Then the history is exponential: $T = T_\infty + (T_0 - T_\infty) e^{-t/\tau}$, where $T_0$ is the starting temperature, $T_\infty$ (T-infinity) is the surrounding fluid's temperature, and $t$ is elapsed time.

The number to carry: after one time constant, 63 % of the excess temperature is GONE and 37 % remains — not the other way round, which is the distractor this lesson keeps setting. Three time constants leaves 5 %, and that is what an engineer means by “cooled”. The companion group is the Fourier number, $\mathrm{Fo} = \dfrac{k t}{\rho c L^{2}}$, dimensionless time for conduction — and the squared length in it is the whole reason heat's reach grows with the square root of the clock rather than with the clock.