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

According to Newton's law of cooling, the rate of heat loss to the surroundings is proportional to the temperature difference between the body and the surroundings.

$\dfrac{dQ}{dt} = k(T - T_0)$

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

Just after the heater is turned off at T = 30^{}^{{}^{\circ}}\text{C}, the rate of cooling is 0.2^{}^{{}^{\circ}}\text{C s}^{-1}. The rate of heat loss at this instant is:

$\dfrac{dQ}{dt} = C \left(-\dfrac{dT}{dt}\right) = 80 \times 0.2 = 16 \text{ W}$

Substituting these values into Newton's law of cooling:

$16 = k(30 - 20)$

k = 1.6 \text{ W }^{}^{{}^{\circ}}\textC^{-1}

The power radiated by the block (rate of heat loss) when its temperature is T = 25^{}^{{}^{\circ}}\text{C} is:

$P = k(25 - 20) = 1.6 \times 5 = 8 \text{ W}$