Question

Newton’s Law of Cooling A cup of coffee is removed from a microwave oven with a temperature of 80°C and allowed to cool in a room with a temperature of 25°C. Five minutes later, the temperature of the coffee is 60°C.
c. When does the temperature of the coffee reach 50°C?

Accepted Answer

Identify the variables and constants in Newton's Law of Cooling. Let $$T(t) be $$the temperature of the coffee at time $$t (in $$minutes), $$T_s = 25^\circ C be $$the surrounding room temperature, and $$T_0 = 80$$^\circ C$$ be the initial temperature of the coffee at t=0$. Write the general form of Newton's Law of Cooling:  
T(t) = T_s + (T_0 - T_s) e$$^{-kt}$$  
where k$ is a positive constant that depends on the cooling rate. Use the given information that after 5 minutes, the temperature is 60°C to find the constant k$. Substitute t=5$, T(5) = 60$, T_s = 25$, and T_0$ = 80$$ into the equation:  
$$60 = 25 + (80 - 25) e^{-5k}$$  
Solve this equation for $$k (do $$not calculate the numerical value, just isolate $$k). $$Set $$T(t) = 50 to $$find the time $$t$$ when the coffee reaches 50°C. Substitute $$T(t) = 50$$, $$T_s = 25$$, $$T_0 = 80$, and the expression for k$ into the formula:  
50 = 25 + (80 - 25) e$$^{-kt}$$ Solve the equation from step 4 for t$ by isolating the exponential term and then taking the natural logarithm. This will give you the time when the coffee reaches 50°C.

Newton’s Law of Cooling A cup of coffee is removed from a microwave oven with a temperature of 80°C and allowed to cool in a room with a temperature of 25°C. Five minutes later, the temperature of the coffee is 60°C.
c. When does the temperature of the coffee reach 50°C?

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