Are the rate of heat emission and the rate of cooling the same? Explain this.

(N/A) No,they are not the same.
$1$. The rate of cooling is defined as the decrease in temperature per unit time $(dT/dt)$. It represents how quickly the temperature of a body drops.
$2$. The rate of heat emission is defined as the loss of heat energy per unit time $(dQ/dt)$. It represents the amount of thermal energy radiated away by the body per second.
$3$. According to the relation $dQ = ms dT$,where $m$ is the mass and $s$ is the specific heat capacity,we have $dQ/dt = ms(dT/dt)$.
$4$. Therefore,the rate of heat emission is proportional to the rate of cooling,but they are physically distinct quantities with different units ($J/s$ for heat emission vs $K/s$ for cooling).