Answer :
To find the estimated value of \( x \) for \( y = 0.049 \), we need to determine the relationship between \( x \) and \( y \) using the provided data. Given the experiment results in the table:
[tex]\[ \begin{array}{|r|c|} \hline x & y \\ \hline 2.5 & 0.400 \\ 9.4 & 0.106 \\ 15.6 & 0.064 \\ 19.5 & 0.051 \\ 25.8 & 0.038 \\ \hline \end{array} \][/tex]
We can see that as \( x \) increases, \( y \) decreases. To estimate \( x \) for \( y = 0.049 \), interpolation between the data points closest to \( y = 0.049 \) will be done.
Looking at the values, we find that \( y = 0.049 \) lies between \( y = 0.051 \) and \( y = 0.038 \):
- For \( x = 19.5 \), \( y = 0.051 \)
- For \( x = 25.8 \), \( y = 0.038 \)
Interpolate these points to find the estimated value of \( x \):
Considering:
\( y_{1} = 0.051 \) \\
\( x_{1} = 19.5 \) \\
\( y_{2} = 0.038 \) \\
\( x_{2} = 25.8 \)
Using linear interpolation:
[tex]\[ x = x_{1} + \frac{(y - y_{1})}{(y_{2} - y_{1})} (x_{2} - x_{1}) \][/tex]
[tex]\[ x = 19.5 + \frac{(0.049 - 0.051)}{(0.038 - 0.051)} (25.8 - 19.5) \][/tex]
Evaluating this step-by-step:
1. Calculate the numerator and denominator for the interpolation fraction.
[tex]\[ y - y_{1} = 0.049 - 0.051 = -0.002 \][/tex]
[tex]\[ y_{2} - y_{1} = 0.038 - 0.051 = -0.013 \][/tex]
2. Compute the fraction:
[tex]\[ \frac{-0.002}{-0.013} = \frac{2}{13} \approx 0.15384615384615385 \][/tex]
3. Apply this fraction to the difference in \( x \) values:
[tex]\[ x_{2} - x_{1} = 25.8 - 19.5 = 6.3 \][/tex]
[tex]\[ 0.15384615384615385 \times 6.3 \approx 0.9692307692307693 \][/tex]
4. Add this value to \( x_{1} \):
[tex]\[ x = 19.5 + 0.9692307692307693 \approx 20.46923076923076 \][/tex]
The estimated value of \( x \) for \( y = 0.049 \) is:
[tex]\[ x \approx 20.469 \][/tex]
Hence, the correct answer to the question is:
A. 20.4
[tex]\[ \begin{array}{|r|c|} \hline x & y \\ \hline 2.5 & 0.400 \\ 9.4 & 0.106 \\ 15.6 & 0.064 \\ 19.5 & 0.051 \\ 25.8 & 0.038 \\ \hline \end{array} \][/tex]
We can see that as \( x \) increases, \( y \) decreases. To estimate \( x \) for \( y = 0.049 \), interpolation between the data points closest to \( y = 0.049 \) will be done.
Looking at the values, we find that \( y = 0.049 \) lies between \( y = 0.051 \) and \( y = 0.038 \):
- For \( x = 19.5 \), \( y = 0.051 \)
- For \( x = 25.8 \), \( y = 0.038 \)
Interpolate these points to find the estimated value of \( x \):
Considering:
\( y_{1} = 0.051 \) \\
\( x_{1} = 19.5 \) \\
\( y_{2} = 0.038 \) \\
\( x_{2} = 25.8 \)
Using linear interpolation:
[tex]\[ x = x_{1} + \frac{(y - y_{1})}{(y_{2} - y_{1})} (x_{2} - x_{1}) \][/tex]
[tex]\[ x = 19.5 + \frac{(0.049 - 0.051)}{(0.038 - 0.051)} (25.8 - 19.5) \][/tex]
Evaluating this step-by-step:
1. Calculate the numerator and denominator for the interpolation fraction.
[tex]\[ y - y_{1} = 0.049 - 0.051 = -0.002 \][/tex]
[tex]\[ y_{2} - y_{1} = 0.038 - 0.051 = -0.013 \][/tex]
2. Compute the fraction:
[tex]\[ \frac{-0.002}{-0.013} = \frac{2}{13} \approx 0.15384615384615385 \][/tex]
3. Apply this fraction to the difference in \( x \) values:
[tex]\[ x_{2} - x_{1} = 25.8 - 19.5 = 6.3 \][/tex]
[tex]\[ 0.15384615384615385 \times 6.3 \approx 0.9692307692307693 \][/tex]
4. Add this value to \( x_{1} \):
[tex]\[ x = 19.5 + 0.9692307692307693 \approx 20.46923076923076 \][/tex]
The estimated value of \( x \) for \( y = 0.049 \) is:
[tex]\[ x \approx 20.469 \][/tex]
Hence, the correct answer to the question is:
A. 20.4
