Solve the problem.
-A cup of coffee is heated to 194° and is then allowed to cool in a room whose air temperature is 72°. After 11 minutes, the temperature of the cup of coffee is 140°. Find the time needed for the coffee to cool to a temperature
Of 102°. Assume the cooling follows Newton's Law of Cooling: $\mathrm { U } = \mathrm { T } + \left( \mathrm { U } _ { \mathrm { O } } - \mathrm { T } \right) \mathrm { e } ^ { \mathrm { kt } }$ (Round your answer to one decimal place.)
A) 15.1 minutes
B) 29.7 minutes
C) 41.1 minutes
D) 26.4 minutes

Question

Quizplus · Accepted Answer

To solve this problem, we first need to find the value of $k$ using the given information. Newton's Law of Cooling is given by $\mathrm { U } = \mathrm { T } + \left( \mathrm { U } _ { \mathrm { O } } - \mathrm { T } \right) \mathrm { e } ^ { \mathrm { kt } }$, where:- $U$ is the final temperature of the object,- $T$ is the ambient temperature,- $U_0$ is the initial temperature of the object,- $k$ is a constant,- $t$ is the time.Given:- Initial temperature, $U_0 = 194°$,- Ambient temperature, $T = 72°$,- Temperature after 11 minutes, $U = 140°$.Substitute these values into the equation to find $k$:$$140 = 72 + (194 - 72)e^{k \cdot 11}$$Solving for $k$, we get a specific value. Then, we use this $k$ value to find the time $t$ when the temperature $U$ is 102°:$$102 = 72 + (194 - 72)e^{k \cdot t}$$Solving for $t$ gives us the time needed for the coffee to cool to 102°, which is approximately 26.4 minutes.

Solve the Problem U=T+(UO−T)ekt\mathrm { U } = \mathrm { T } + \left( \mathrm { U } _ { \mathrm { O } } - \mathrm { T } \right) \mathrm { e } ^ { \mathrm { kt } }U=T+(UO​−T)ekt

Solve the Problem $\mathrm { U } = \mathrm { T } + \left( \mathrm { U } _ { \mathrm { O } } - \mathrm { T } \right) \mathrm { e } ^ { \mathrm { kt } }$