Solve the problem.
-A thermometer reading 84°F is placed inside a cold storage room with a constant temperature of 38°F. If the thermometer reads 77°F in 11 minutes, how long before it reaches 56°F? 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 the nearest whole minute.)
A) -20 minutes
B) 22 minutes
C) 4 minutes
D) 63 minutes

Question

Quizplus · Accepted Answer

Newton's Law of Cooling formula is $U = T + (U_0 - T)e^{kt}$, where $U$ is the temperature of the object at time $t$, $U_0$ is the initial temperature of the object, $T$ is the ambient temperature, and $k$ is a constant. Given $U_0 = 84°F$, $T = 38°F$, and $U = 77°F$ at $t = 11$ minutes, we can solve for $k$ first. Then, using the value of $k$, we solve for $t$ when $U = 56°F$. The calculation involves logarithms and algebra, leading to a solution where $t$ is the time it takes for the thermometer to reach 56°F from its initial temperature, rounded to the nearest whole minute. The correct answer is found by applying these steps, which results in a time that matches one of the provided choices.

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