Final answer:
To find the coffee's temperature after 20 minutes, we can use Newton's Law of Cooling. The temperature of the coffee after 20 minutes is approximately 50.8°F.
Step-by-step explanation:
To find the coffee's temperature after 20 minutes, we can use Newton's Law of Cooling. Newton's Law of Cooling states that the rate of change of temperature of an object is proportional to the difference in temperature between the object and its surroundings. The equation for Newton's Law of Cooling is:
T(t) = T_s + (T_0 - T_s)e^(-kt)
Where:
- T(t) is the temperature of the object at time t
- T_s is the temperature of the surroundings
- T_0 is the initial temperature of the object
- k is a constant that depends on the cooling properties of the object
In this case, the initial temperature of the coffee is 160°F, the temperature of the surroundings is 0°F, and after 15 minutes, the temperature of the coffee is 47°F. Plugging these values into the equation, we can solve for k:
47 = 0 + (160 - 0)e^(-15k)
After solving this equation, we find that k is approximately 0.025. Now we can use this value of k to find the temperature of the coffee after 20 minutes:
T(20) = 0 + (160 - 0)e^(-20*0.025)
Calculating this expression, we find that T(20) is approximately 50.8°F.