To estimate the value of [tex]\( x \)[/tex] for [tex]\( y = 0.049 \)[/tex], we can use linear interpolation between the given data points. Here's a detailed step-by-step solution:
1. Identify the interval for interpolation:
We need to find the interval in which [tex]\( y = 0.049 \)[/tex] falls. Looking at our dataset:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
2.5 & 0.400 \\
\hline
9.4 & 0.106 \\
\hline
15.6 & 0.064 \\
\hline
19.5 & 0.051 \\
\hline
25.8 & 0.038 \\
\hline
\end{array}
\][/tex]
We observe that [tex]\( y = 0.049 \)[/tex] falls between 0.051 (at [tex]\( x = 19.5 \)[/tex]) and 0.038 (at [tex]\( x = 25.8 \)[/tex]).
2. Identify the endpoints for [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
The two points for interpolation are:
- Point 1: [tex]\((x_1 = 19.5, y_1 = 0.051)\)[/tex]
- Point 2: [tex]\((x_2 = 25.8, y_2 = 0.038)\)[/tex]
3. Apply the linear interpolation formula:
The linear interpolation formula to estimate [tex]\( x \)[/tex] is:
[tex]\[
x = x_1 + \frac{(y - y_1) \cdot (x_2 - x_1)}{(y_2 - y_1)}
\][/tex]
Substituting the values:
[tex]\[
x = 19.5 + \frac{(0.049 - 0.051) \cdot (25.8 - 19.5)}{(0.038 - 0.051)}
\][/tex]
4. Simplify the calculations:
[tex]\[
x = 19.5 + \frac{-0.002 \cdot 6.3}{-0.013}
\][/tex]
[tex]\[
x = 19.5 + \frac{0.0126}{0.013}
\][/tex]
[tex]\[
x = 19.5 + 0.9692307692307692
\][/tex]
[tex]\[
x \approx 20.469230769230766
\][/tex]
Given the result, the correct answer is:
A. 20.4
