Question

Suppose you have just poured a cup of freshly brewed coffee with temperature 90∘C in a room where the temperature is 20∘C. Newton's Law of Cooling states that the rate of cooling of an object is proportional to the temperature difference between the object and its surroundings. Therefore, the temperature of the coffee, T(t), satisfies the differential equation dTdt=k(T−Troom) where Troom=20 is the room temperature, and k is some constant. Suppose it is known that the coffee cools at a rate of 2∘C per minute when its temperature is 60∘C. A. What is the limiting value of the temperature of the coffee?

Answer 1

The limiting value of the temperature of the coffee is 20°C. We get Tlimit - 20 = 0, and solving for Tlimit gives us Tlimit = 20°C. Therefore, the limiting value of the temperature of the coffee is 20°C.

Step-by-step explanation:

Let's solve the given differential equation to find the limiting value of the temperature of the coffee.

The given differential equation is dT/dt = k(T - Troom), where Troom = 20°C and k is a constant. We are given that the coffee cools at a rate of 2°C per minute when its temperature is 60°C.

To find the limiting value, we can set dT/dt to zero and solve for T.

So, 0 = k(Tlimit - Troom). Plugging in the given values, we get 0 = k(Tlimit - 20) and since k is nonzero, we can divide both sides by k.

We get Tlimit - 20 = 0, and solving for Tlimit gives us Tlimit = 20°C. Therefore, the limiting value of the temperature of the coffee is 20°C.

Learn more about Newton's Law of Cooling