An object cools at a rate (measured in ${ } ^ { \circ } \mathrm { C } / \mathrm { min }$ ) equal to $k$ times the difference between its temperature and that of the surrounding air. Suppose the object takes 10 minutes to cool from 60 $^\circ$ C to 40 $^\circ$ C in a room kept at 20 $^\circ$ C. Find the value of $k$ .
A) $e ^ { - 20 }$
B) $\ln 2$
C) $10 e ^ { - 20 }$
D) $40 \ln 10$
E) $\frac { 1 } { 2 }$
F) $e ^ { - 1 / 20 }$
G) $\frac { 1 } { 10 } \ln \frac { 1 } { 2 }$
H) $60 \ln \frac { 1 } { 2 }$

Question

An object cools at a rate (measured in ${ } ^ { \circ } \mathrm { C } / \mathrm { min }$ ) equal to $k$ times the difference between its temperature and that of the surrounding air. Suppose the object takes 10 minutes to cool from 60 $^\circ$ C to 40 $^\circ$ C in a room kept at 20 $^\circ$ C. Find the value of $k$ .
A) $e ^ { - 20 }$ 
B) $\ln 2$ 
C) $10 e ^ { - 20 }$ 
D) $40 \ln 10$
E) $\frac { 1 } { 2 }$ 
F) $e ^ { - 1 / 20 }$ 
G) $\frac { 1 } { 10 } \ln \frac { 1 } { 2 }$ 
H) $60 \ln \frac { 1 } { 2 }$

Quizplus · Accepted Answer

The cooling process is described by Newton's Law of Cooling, which states that the rate of change of the temperature of an object is proportional to the difference between its temperature and the ambient temperature. The formula for the temperature $T(t)$ of an object at time $t$ in this context is given by $T(t) = T_{\text{ambient}} + (T_{\text{initial}} - T_{\text{ambient}})e^{-kt}$, where $T_{\text{ambient}}$ is the ambient temperature, $T_{\text{initial}}$ is the initial temperature of the object, and $k$ is the constant of proportionality.Given that the object cools from 60°C to 40°C in 10 minutes in a room at 20°C, we can set up the equation $40 = 20 + (60 - 20)e^{-10k}$. Simplifying, we get $20 = 40e^{-10k}$, or $1/2 = e^{-10k}$. Taking the natural logarithm of both sides gives $\ln(1/2) = -10k$, so $k = \frac{1}{10}\ln(1/2)$, which corresponds to choice G.

An Object Cools at a Rate (Measured In ∘C/min{ } ^ { \circ } \mathrm { C } / \mathrm { min }∘C/min

An Object Cools at a Rate (Measured In ${ } ^ { \circ } \mathrm { C } / \mathrm { min }$