A metal sphere cools at a rate of 2^ ^ C/min when the temperature...

According to Newton's law of cooling, the rate of cooling is directly proportional to the temperature difference between the object and its surroundings.

$\dfrac{dT}{dt} = k(T - T_s)$

Given that the rate of cooling is 2^{}^{\circ}\text{C/min} when T = 80^{}^{\circ}\text{C} and T_s = 30^{}^{\circ}\text{C}

$2 = k(80 - 30)$

$2 = 50k \Rightarrow k = \dfrac{2}{50} = \dfrac{1}{25}$

When the temperature of the sphere is 50^{}^{\circ}\text{C}, the new rate of cooling is

$\dfrac{dT}{dt} = k(50 - 30)$

\dfrac{dT}{dt} = \dfrac{1}{25} \times 20 = \dfrac{4}{5} = 0.8^{}^{\circ}\text{C/min}

Answer: 0.8^{}^{\circ}\text{C/min}