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Answer :
To determine the time it takes for the oven to cool to a specific temperature using the function [tex]\( f(t) = 349.2 \times (0.98)^t \)[/tex], we need to find the value of [tex]\( t \)[/tex] when the oven reaches a temperature of 100 degrees Fahrenheit.
Here’s how we can find that:
1. Set up the equation:
You want to find [tex]\( t \)[/tex] when the temperature [tex]\( f(t) \)[/tex] is 100. So, you set up the equation:
[tex]\[
f(t) = 349.2 \times (0.98)^t = 100
\][/tex]
2. Solve for [tex]\( t \)[/tex]:
To solve for [tex]\( t \)[/tex], first divide both sides of the equation by 349.2:
[tex]\[
(0.98)^t = \frac{100}{349.2}
\][/tex]
3. Use logarithms to solve for [tex]\( t \)[/tex]:
Take the natural logarithm on both sides to bring down the exponent:
[tex]\[
\ln((0.98)^t) = \ln\left(\frac{100}{349.2}\right)
\][/tex]
4. Apply the power rule for logarithms:
Using the property of logarithms [tex]\(\ln(a^b) = b \ln(a)\)[/tex], the equation becomes:
[tex]\[
t \cdot \ln(0.98) = \ln\left(\frac{100}{349.2}\right)
\][/tex]
5. Solve for [tex]\( t \)[/tex]:
Isolate [tex]\( t \)[/tex] by dividing both sides by [tex]\(\ln(0.98)\)[/tex]:
[tex]\[
t = \frac{\ln\left(\frac{100}{349.2}\right)}{\ln(0.98)}
\][/tex]
6. Calculate the value of [tex]\( t \)[/tex]:
After performing the calculations, you find that [tex]\( t \)[/tex] approximately equals 62 minutes.
Therefore, the time it will take for the oven to cool to 100 degrees Fahrenheit is approximately 62 minutes.
Here’s how we can find that:
1. Set up the equation:
You want to find [tex]\( t \)[/tex] when the temperature [tex]\( f(t) \)[/tex] is 100. So, you set up the equation:
[tex]\[
f(t) = 349.2 \times (0.98)^t = 100
\][/tex]
2. Solve for [tex]\( t \)[/tex]:
To solve for [tex]\( t \)[/tex], first divide both sides of the equation by 349.2:
[tex]\[
(0.98)^t = \frac{100}{349.2}
\][/tex]
3. Use logarithms to solve for [tex]\( t \)[/tex]:
Take the natural logarithm on both sides to bring down the exponent:
[tex]\[
\ln((0.98)^t) = \ln\left(\frac{100}{349.2}\right)
\][/tex]
4. Apply the power rule for logarithms:
Using the property of logarithms [tex]\(\ln(a^b) = b \ln(a)\)[/tex], the equation becomes:
[tex]\[
t \cdot \ln(0.98) = \ln\left(\frac{100}{349.2}\right)
\][/tex]
5. Solve for [tex]\( t \)[/tex]:
Isolate [tex]\( t \)[/tex] by dividing both sides by [tex]\(\ln(0.98)\)[/tex]:
[tex]\[
t = \frac{\ln\left(\frac{100}{349.2}\right)}{\ln(0.98)}
\][/tex]
6. Calculate the value of [tex]\( t \)[/tex]:
After performing the calculations, you find that [tex]\( t \)[/tex] approximately equals 62 minutes.
Therefore, the time it will take for the oven to cool to 100 degrees Fahrenheit is approximately 62 minutes.
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