Answer :
To find the constants [tex]\( p \)[/tex] and [tex]\( q \)[/tex] in the relation [tex]\(\left(\frac{s}{t}\right)^p = q e^{-t}\)[/tex], we will transform the equation and use a linear regression approach on logarithmic scales. Here’s a step-by-step outline to solve this problem:
1. Transform the Relation:
Start with the given equation:
[tex]\[ \left(\frac{s}{t}\right)^p = q e^{-t} \][/tex]
Take the natural logarithm of both sides:
[tex]\[ \ln\left(\left(\frac{s}{t}\right)^p\right) = \ln \left( q e^{-t} \right) \][/tex]
Apply logarithm properties:
[tex]\[ p \cdot \ln \left( \frac{s}{t} \right) = \ln(q) - t \][/tex]
Simplify further:
[tex]\[ p \left( \ln(s) - \ln(t) \right) = \ln(q) - t \][/tex]
2. Prepare the Data for Linear Regression:
Let's list the values of [tex]\( t \)[/tex] and [tex]\( s \)[/tex] from the provided table.
[tex]\[ \begin{array}{|c|c|c|c|c|c|} \hline t & 0.2 & 0.4 & 0.6 & 0.8 & 1.0 \\ \hline s & 1.09 & 1.96 & 2.67 & 3.22 & 3.64 \\ \hline \end{array} \][/tex]
3. Compute [tex]\(\ln(s)\)[/tex] and [tex]\(\ln(t)\)[/tex] for Each Pair ([tex]\(t, s\)[/tex]):
[tex]\[ \begin{array}{|c|c|c|} \hline t & s & \ln(s) & \ln(t) \\ \hline 0.2 & 1.09 & 0.0862 & -1.6094 \\ \hline 0.4 & 1.96 & 0.6729 & -0.9163 \\ \hline 0.6 & 2.67 & 0.9821 & -0.5108 \\ \hline 0.8 & 3.22 & 1.1694 & -0.2231 \\ \hline 1.0 & 3.64 & 1.2920 & 0.0000 \\ \hline \end{array} \][/tex]
4. Perform Linear Regression on [tex]\( \ln(s) \)[/tex] vs [tex]\( \ln(t) \)[/tex]:
We assume:
[tex]\[ y = \ln(s) \quad \text{and} \quad x = \ln(t) \][/tex]
The linear relation [tex]\( y = mx + b \)[/tex] can be determined by fitting a line to the data points [tex]\((\ln(t), \ln(s))\)[/tex].
From the previous calculation:
- The slope ([tex]\( m \)[/tex]) of this linear regression is 0.7564.
- The intercept ([tex]\( b \)[/tex]) of this linear regression is 1.3336.
5. Calculate [tex]\( p \)[/tex] and [tex]\( q \)[/tex]:
- The slope ([tex]\( m \)[/tex]) corresponds to the exponent [tex]\( p \)[/tex]:
[tex]\[ p = 0.7564 \][/tex]
- The intercept ([tex]\( b \)[/tex]) corresponds to [tex]\( \ln(q) \)[/tex]:
[tex]\[ \ln(q) = 1.3336 \implies q = e^{1.3336} = 3.7947 \][/tex]
Finally, the values of the constants are:
[tex]\[ p \approx 0.756 \quad \text{and} \quad q \approx 3.795 \][/tex]
Thus, the relationship between [tex]\( s \)[/tex] and [tex]\( t \)[/tex] is successfully determined.
1. Transform the Relation:
Start with the given equation:
[tex]\[ \left(\frac{s}{t}\right)^p = q e^{-t} \][/tex]
Take the natural logarithm of both sides:
[tex]\[ \ln\left(\left(\frac{s}{t}\right)^p\right) = \ln \left( q e^{-t} \right) \][/tex]
Apply logarithm properties:
[tex]\[ p \cdot \ln \left( \frac{s}{t} \right) = \ln(q) - t \][/tex]
Simplify further:
[tex]\[ p \left( \ln(s) - \ln(t) \right) = \ln(q) - t \][/tex]
2. Prepare the Data for Linear Regression:
Let's list the values of [tex]\( t \)[/tex] and [tex]\( s \)[/tex] from the provided table.
[tex]\[ \begin{array}{|c|c|c|c|c|c|} \hline t & 0.2 & 0.4 & 0.6 & 0.8 & 1.0 \\ \hline s & 1.09 & 1.96 & 2.67 & 3.22 & 3.64 \\ \hline \end{array} \][/tex]
3. Compute [tex]\(\ln(s)\)[/tex] and [tex]\(\ln(t)\)[/tex] for Each Pair ([tex]\(t, s\)[/tex]):
[tex]\[ \begin{array}{|c|c|c|} \hline t & s & \ln(s) & \ln(t) \\ \hline 0.2 & 1.09 & 0.0862 & -1.6094 \\ \hline 0.4 & 1.96 & 0.6729 & -0.9163 \\ \hline 0.6 & 2.67 & 0.9821 & -0.5108 \\ \hline 0.8 & 3.22 & 1.1694 & -0.2231 \\ \hline 1.0 & 3.64 & 1.2920 & 0.0000 \\ \hline \end{array} \][/tex]
4. Perform Linear Regression on [tex]\( \ln(s) \)[/tex] vs [tex]\( \ln(t) \)[/tex]:
We assume:
[tex]\[ y = \ln(s) \quad \text{and} \quad x = \ln(t) \][/tex]
The linear relation [tex]\( y = mx + b \)[/tex] can be determined by fitting a line to the data points [tex]\((\ln(t), \ln(s))\)[/tex].
From the previous calculation:
- The slope ([tex]\( m \)[/tex]) of this linear regression is 0.7564.
- The intercept ([tex]\( b \)[/tex]) of this linear regression is 1.3336.
5. Calculate [tex]\( p \)[/tex] and [tex]\( q \)[/tex]:
- The slope ([tex]\( m \)[/tex]) corresponds to the exponent [tex]\( p \)[/tex]:
[tex]\[ p = 0.7564 \][/tex]
- The intercept ([tex]\( b \)[/tex]) corresponds to [tex]\( \ln(q) \)[/tex]:
[tex]\[ \ln(q) = 1.3336 \implies q = e^{1.3336} = 3.7947 \][/tex]
Finally, the values of the constants are:
[tex]\[ p \approx 0.756 \quad \text{and} \quad q \approx 3.795 \][/tex]
Thus, the relationship between [tex]\( s \)[/tex] and [tex]\( t \)[/tex] is successfully determined.
