To determine how many minutes Amy should leave the bottle of water in the cooler until it reaches a temperature of 21 degrees Celsius, we need to solve the given temperature function for [tex]\( t \)[/tex]. The temperature function is given by:
[tex]\[ C(t) = 4 + 20 e^{-0.05t} \][/tex]
We want to find [tex]\( t \)[/tex] when [tex]\( C(t) = 21 \)[/tex]. So, we set up the equation:
[tex]\[ 21 = 4 + 20 e^{-0.05t} \][/tex]
Next, we solve for [tex]\( t \)[/tex]:
1. Subtract 4 from both sides to isolate the exponential term:
[tex]\[ 21 - 4 = 20 e^{-0.05t} \][/tex]
[tex]\[ 17 = 20 e^{-0.05t} \][/tex]
2. Divide both sides by 20 to further isolate the exponential term:
[tex]\[ \frac{17}{20} = e^{-0.05t} \][/tex]
3. Take the natural logarithm of both sides to solve for the exponent:
[tex]\[ \ln\left(\frac{17}{20}\right) = \ln\left(e^{-0.05t}\right) \][/tex]
[tex]\[ \ln\left(\frac{17}{20}\right) = -0.05t \][/tex]
4. Rearrange to solve for [tex]\( t \)[/tex]:
[tex]\[ t = \frac{\ln\left(\frac{17}{20}\right)}{-0.05} \][/tex]
Using the calculated natural logarithm and performing the division, we find:
[tex]\[ t \approx 3.250378589955499 \][/tex]
Therefore, Amy should leave the bottle in the cooler for approximately [tex]\( 3.25 \)[/tex] minutes until it reaches a temperature of 21 degrees Celsius.
