To determine how long Miguel should leave the water in the cooler until it reaches a temperature of 15 degrees Celsius, we need to solve for [tex]\( t \)[/tex] in the given exponential function [tex]\( C(t) = 8 + 14e^{-0.028t} \)[/tex].
Here are the steps involved:
1. Set the temperature equation equal to 15 degrees:
[tex]\[
15 = 8 + 14e^{-0.028t}
\][/tex]
2. Isolate the exponential term:
To isolate [tex]\( e^{-0.028t} \)[/tex], subtract 8 from both sides:
[tex]\[
15 - 8 = 14e^{-0.028t}
\][/tex]
Simplify this to:
[tex]\[
7 = 14e^{-0.028t}
\][/tex]
3. Solve for the exponential term:
Divide both sides by 14:
[tex]\[
\frac{7}{14} = e^{-0.028t}
\][/tex]
Simplify the fraction:
[tex]\[
0.5 = e^{-0.028t}
\][/tex]
4. Take the natural logarithm of both sides:
To solve for [tex]\( t \)[/tex], take the natural logarithm ([tex]\(\ln\)[/tex]) of both sides:
[tex]\[
\ln(0.5) = \ln(e^{-0.028t})
\][/tex]
5. Use the property of logarithms:
Apply the property [tex]\(\ln(e^x) = x\)[/tex]:
[tex]\[
\ln(0.5) = -0.028t
\][/tex]
6. Solve for [tex]\( t \)[/tex]:
Divide both sides by -0.028:
[tex]\[
t = \frac{\ln(0.5)}{-0.028}
\][/tex]
7. Calculate the value:
Perform the calculation:
[tex]\[
t \approx \frac{-0.6931}{-0.028} \approx 24.75525644856948
\][/tex]
8. Round to the nearest tenth:
To round this to the nearest tenth:
[tex]\[
t \approx 24.8
\][/tex]
Thus, Miguel should leave the water in the cooler for approximately 24.8 minutes to reach a temperature of 15 degrees Celsius.
