To solve this problem, we need to find the value of the decay constant [tex]\( k \)[/tex] using the given data and then use this value to determine the temperature of the can of soda after a specified time.
Given data:
- Initial temperature of the soda, [tex]\( T_0 = 70^\circ F \)[/tex]
- Temperature of the refrigerator, [tex]\( T_a = 34^\circ F \)[/tex]
- Temperature of the soda after 40 minutes, [tex]\( T = 58^\circ F \)[/tex]
- Time, [tex]\( t = 40 \)[/tex] minutes
We use the formula provided by Newton's Law of Cooling:
[tex]\[
T = T_a + (T_0 - T_a) e^{-k t}
\][/tex]
1. Finding the decay constant [tex]\( k \)[/tex]:
- Plug in the known values:
[tex]\[
58 = 34 + (70 - 34) e^{-k \cdot 40}
\][/tex]
- Simplify inside the parenthesis:
[tex]\[
58 = 34 + 36 e^{-k \cdot 40}
\][/tex]
- Subtract 34 from both sides to isolate the exponential term:
[tex]\[
24 = 36 e^{-k \cdot 40}
\][/tex]
- Divide both sides by 36 to further isolate the exponential term:
[tex]\[
\frac{24}{36} = e^{-k \cdot 40}
\][/tex]
[tex]\[
\frac{2}{3} = e^{-k \cdot 40}
\][/tex]
- Take the natural logarithm of both sides to solve for [tex]\( k \)[/tex]:
[tex]\[
\ln\left(\frac{2}{3}\right) = -k \cdot 40
\][/tex]
- Isolate [tex]\( k \)[/tex] by dividing both sides by [tex]\(-40\)[/tex]:
[tex]\[
k = -\frac{\ln\left(\frac{2}{3}\right)}{40}
\][/tex]
- Simplify to find [tex]\( k \)[/tex]:
[tex]\[
k \approx 0.01
\][/tex]
2. Determining the temperature of the can of soda after 60 minutes:
- Using [tex]\( t = 60 \)[/tex] minutes, we use the formula again:
[tex]\[
T = T_a + (T_0 - T_a) e^{-k t}
\][/tex]
[tex]\[
T = 34 + (70 - 34) e^{-0.01 \cdot 60}
\][/tex]
[tex]\[
T = 34 + 36 e^{-0.6}
\][/tex]
- Calculate the exponential term and then the temperature [tex]\( T \)[/tex]:
[tex]\[
e^{-0.6} \approx 0.549
\][/tex]
[tex]\[
T \approx 34 + 36 \cdot 0.549
\][/tex]
[tex]\[
T \approx 34 + 19.764
\][/tex]
[tex]\[
T \approx 53.764
\][/tex]
- Round to the nearest integer:
[tex]\[
T \approx 54^\circ F
\][/tex]
Therefore, the value of [tex]\( k \)[/tex] is [tex]\( 0.01 \)[/tex] and the temperature of the can of soda after 60 minutes in the refrigerator is [tex]\( 54^\circ F \)[/tex].
