We appreciate your visit to An object has been heated to 190 degrees Fahrenheit and is brought into a room where the temperature is 65 degrees Fahrenheit After 2 minutes. This page offers clear insights and highlights the essential aspects of the topic. Our goal is to provide a helpful and engaging learning experience. Explore the content and find the answers you need!
Answer :
To solve this problem, we need to model the change in temperature of the object using Newton's Law of Cooling. This law states that the rate of change of the temperature of an object is proportional to the difference between its temperature and the ambient temperature.
### Step-by-Step Solution:
1. Identify key values:
- Initial temperature of the object, [tex]\( T_0 = 190^\circ \text{F} \)[/tex].
- Ambient room temperature, [tex]\( T_{\text{ambient}} = 65^\circ \text{F} \)[/tex].
- Temperature of the object after 2 minutes, [tex]\( T(2) = 150^\circ \text{F} \)[/tex].
2. Newton's Law of Cooling formula:
- The equation for Newton's Law of Cooling is:
[tex]\[
T(t) = T_{\text{ambient}} + (T_0 - T_{\text{ambient}}) \cdot e^{-kt}
\][/tex]
where [tex]\( T(t) \)[/tex] is the temperature at time [tex]\( t \)[/tex], and [tex]\( k \)[/tex] is the cooling rate constant.
3. Set up the equation for [tex]\( T(2) \)[/tex]:
- We know [tex]\( T(2) = 150^\circ \text{F} \)[/tex].
- Substituting into the formula:
[tex]\[
150 = 65 + (190 - 65) \cdot e^{-2k}
\][/tex]
4. Solve for [tex]\( k \)[/tex]:
- Rearrange the equation:
[tex]\[
150 - 65 = 125 \cdot e^{-2k}
\][/tex]
[tex]\[
85 = 125 \cdot e^{-2k}
\][/tex]
[tex]\[
\frac{85}{125} = e^{-2k}
\][/tex]
[tex]\[
e^{-2k} = 0.68
\][/tex]
5. Calculate [tex]\( k \)[/tex]:
- Take the natural logarithm of both sides:
[tex]\[
-2k = \ln(0.68)
\][/tex]
- Solving for [tex]\( k \)[/tex]:
[tex]\[
k = -\frac{\ln(0.68)}{2}
\][/tex]
- From the given answer, this value evaluates to approximately [tex]\( k \approx 0.1928 \)[/tex].
6. Write the function:
- Substitute [tex]\( k \)[/tex] back into the original equation:
[tex]\[
T(t) = 65 + 125 \cdot e^{-0.1928t}
\][/tex]
Therefore, the equation that models the temperature of the object as a function of time [tex]\( t \)[/tex] is:
[tex]\[ T(t) = 65 + 125 \cdot e^{-0.1928t} \][/tex]
### Step-by-Step Solution:
1. Identify key values:
- Initial temperature of the object, [tex]\( T_0 = 190^\circ \text{F} \)[/tex].
- Ambient room temperature, [tex]\( T_{\text{ambient}} = 65^\circ \text{F} \)[/tex].
- Temperature of the object after 2 minutes, [tex]\( T(2) = 150^\circ \text{F} \)[/tex].
2. Newton's Law of Cooling formula:
- The equation for Newton's Law of Cooling is:
[tex]\[
T(t) = T_{\text{ambient}} + (T_0 - T_{\text{ambient}}) \cdot e^{-kt}
\][/tex]
where [tex]\( T(t) \)[/tex] is the temperature at time [tex]\( t \)[/tex], and [tex]\( k \)[/tex] is the cooling rate constant.
3. Set up the equation for [tex]\( T(2) \)[/tex]:
- We know [tex]\( T(2) = 150^\circ \text{F} \)[/tex].
- Substituting into the formula:
[tex]\[
150 = 65 + (190 - 65) \cdot e^{-2k}
\][/tex]
4. Solve for [tex]\( k \)[/tex]:
- Rearrange the equation:
[tex]\[
150 - 65 = 125 \cdot e^{-2k}
\][/tex]
[tex]\[
85 = 125 \cdot e^{-2k}
\][/tex]
[tex]\[
\frac{85}{125} = e^{-2k}
\][/tex]
[tex]\[
e^{-2k} = 0.68
\][/tex]
5. Calculate [tex]\( k \)[/tex]:
- Take the natural logarithm of both sides:
[tex]\[
-2k = \ln(0.68)
\][/tex]
- Solving for [tex]\( k \)[/tex]:
[tex]\[
k = -\frac{\ln(0.68)}{2}
\][/tex]
- From the given answer, this value evaluates to approximately [tex]\( k \approx 0.1928 \)[/tex].
6. Write the function:
- Substitute [tex]\( k \)[/tex] back into the original equation:
[tex]\[
T(t) = 65 + 125 \cdot e^{-0.1928t}
\][/tex]
Therefore, the equation that models the temperature of the object as a function of time [tex]\( t \)[/tex] is:
[tex]\[ T(t) = 65 + 125 \cdot e^{-0.1928t} \][/tex]
Thanks for taking the time to read An object has been heated to 190 degrees Fahrenheit and is brought into a room where the temperature is 65 degrees Fahrenheit After 2 minutes. We hope the insights shared have been valuable and enhanced your understanding of the topic. Don�t hesitate to browse our website for more informative and engaging content!
- Why do Businesses Exist Why does Starbucks Exist What Service does Starbucks Provide Really what is their product.
- The pattern of numbers below is an arithmetic sequence tex 14 24 34 44 54 ldots tex Which statement describes the recursive function used to..
- Morgan felt the need to streamline Edison Electric What changes did Morgan make.
Rewritten by : Barada
