A metal block having a heat capacity of 80 J ^ ^ ^ ^ -1 is located...

Let the power of the heater be $P$ and the cooling constant be $k$.

According to Newton's law of cooling, the rate of change of temperature of the block when the heater is on is given by:

$C \dfrac{dT}{dt} = P - k(T - T_0)$

where C = 80 \text{ J }^{}^{{}^{\circ}}\textC^{-1} is the heat capacity and T_0 = 20^{}^{{}^{\circ}}\text{C} is the room temperature.

Just after the heater is turned on, the temperature of the block is equal to the room temperature, T = 20^{}^{{}^{\circ}}\text{C}. Since $T = T_0$, the rate of heat loss to the surroundings is zero.

Substituting the given rate of heating \dfrac{dT}{dt} = 2^{}^{{}^{\circ}}\text{C s}^{-1} at T = 20^{}^{{}^{\circ}}\text{C}:

$80 \times 2 = P - k(20 - 20)$

$P = 160 \text{ W}$

The data provided for the cooling rate at 30^{}^{{}^{\circ}}\text{C} is consistent but not required to determine the power of the heater.