Suppose you have just poured a cup of freshly brewed coffee with temperature 95

Question

Suppose you have just poured a cup of freshly brewed coffee with temperature 95 degree C in a room where the temperature is 25 degree 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 dT/dt = k(T - T_room) where T_room = 25 is the room temperature, and k is some constant. Suppose it is known that the coffee cools at a rate of 2 degree C per minute when its temperature is 65 degree C. What is the limiting value of the temperature of the coffee? lim What is the limiting value of the rate of cooling? [math] Find the constant [math] in the differential equation. [math] Use Euler's method with step size [math] minutes to estimate the temperature of the coffee after [math] minutes. [math]

Explanation / Answer

A) Limiting value of the temperature = 25°C

B)

limiting value of dT/dt = k[(limit temp) - (room temp)]

                                  = k(25 - 25)

                                  = 0

C)

dT/dt = k(T - 25)

-2 = k (65 - 25)

-2 = 40k

k = -0.05

D)

T' = -0.05 x (95-25) = - 3.5

h = 2, deltaT = -3.5x 2 = -7

T = 95-7 = 88

Then T' = -0.05 x (88-25) = - 3.15

T = 88 - 2 x 3.15 = 81.7

81.7 after 4 minutes.

Then T' = -0.05x (81.7-25) = -2.835

T = 81.7 - 2x 2.835 = 76.03 after 6 mins

Then T' = -0.05 x (76.03-25) = -2.5515

T = 76.03 - 2 x 2.5515 = 70.92700 after 8 mins

Then T' = -0.05 x (70.927-25) = -2.29635

T = 70.927 - 2 x 2.29635 = 66.3343 after 10 mins

so

temperature = 66.33 after 10 mins by eulars method

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