Question #42141

A nuclear micro-reactor powers a heating system by converting mass into energy. The rate of mass defect in the reactor varies with time as $\frac{dm}{dt} = \lambda t$, where $\lambda = 1.4 \times 10^{-13} \text{ kg/s}^2$. The reactor operates from $t = 0$ to $t = 10 \text{ s}$. The entire energy released is used to heat a mixture of two liquids, A and B, kept in a perfectly insulated container. Liquid A has a mass of $1 \text{ kg}$ and a temperature-dependent specific heat capacity $s_A(T) = \alpha T^2$, where $\alpha = 0.01 \text{ J kg}^{-1} \text{K}^{-3}$. Liquid B has a mass of $2 \text{ kg}$ and a specific heat capacity $s_B(T) = \alpha T^2$. If the initial temperature of the mixture is $100 \text{ K}$, find the final temperature of the mixture. (Take the speed of light $c = 3 \times 10^8 \text{ m/s}$).