Choose the one alternative that best completes the statement or answers the question.
-Sandy manages a ceramics shop and uses a $650 ^ { \circ } \mathrm { F }$ kiln to fire ceramic greenware. After turning off her kiln, she must wait until its temperature gauge reaches $205 ^ { \circ } \mathrm { F }$ before opening it and removing the ceramic pieces. If room temperature is $70 ^ { \circ } \mathrm { F }$ and the gauge reads $500 ^ { \circ } \mathrm { F }$ in 8 minutes, how long must she wait before opening the kiln? Assume the kiln cools according to 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) 79 minutes
B) 56 minutes
C) 30 minutes
D) 39 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 that the kiln cools from $650^\circ F$ to $500^\circ F$ in 8 minutes with an ambient temperature of $70^\circ F$, we can solve for $k$ and then use this value to find the time it takes to cool to $205^\circ F$. The calculation involves setting up the equation with the given values and solving for $t$ when $U = 205$. The correct answer, after performing these calculations and rounding to the nearest whole minute, is 39 minutes.

Choose the One Alternative That Best Completes the Statement or Answers