At Westonci.ca, we provide reliable answers to your questions from a community of experts. Start exploring today! Get accurate and detailed answers to your questions from a dedicated community of experts on our Q&A platform. Our platform provides a seamless experience for finding reliable answers from a network of experienced professionals.
Sagot :
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.
[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.
We appreciate your time. Please revisit us for more reliable answers to any questions you may have. We appreciate your time. Please come back anytime for the latest information and answers to your questions. We're dedicated to helping you find the answers you need at Westonci.ca. Don't hesitate to return for more.