To estimate the value of [tex]\( y \)[/tex] for [tex]\( x = 4 \)[/tex] using the given data points, we will use interpolation. Interpolation is a method of estimating unknown values that fall between known values.
Given the data points:
[tex]\[
\begin{array}{|c|c|}
\hline x & y \\
\hline 2.5 & 6.25 \\
\hline 9.4 & 88.36 \\
\hline 15.6 & 243.63 \\
\hline 19.5 & 380.25 \\
\hline 25.8 & 665.64 \\
\hline
\end{array}
\][/tex]
We need to find [tex]\( y \)[/tex] for [tex]\( x = 4 \)[/tex].
The values of [tex]\( x \)[/tex] given are 2.5 and 9.4, among others. Since 4 is between 2.5 and 9.4, we will look at these two points for interpolation:
[tex]\[
\begin{align*}
x_1 &= 2.5, & y_1 &= 6.25 \\
x_2 &= 9.4, & y_2 &= 88.36
\end{align*}
\][/tex]
Linear interpolation involves finding the linear function between these two points and then using it to estimate the value of [tex]\( y \)[/tex] for [tex]\( x = 4 \)[/tex].
The formula for linear interpolation is:
[tex]\[
y = y_1 + \frac{(y_2 - y_1)}{(x_2 - x_1)} \cdot (x - x_1)
\][/tex]
Substitute the values:
[tex]\[
\begin{align*}
y &= 6.25 + \frac{(88.36 - 6.25)}{(9.4 - 2.5)} \cdot (4 - 2.5) \\
&= 6.25 + \frac{(82.11)}{(6.9)} \cdot 1.5 \\
&= 6.25 + 11.903 \cdot 1.5 \\
&= 6.25 + 17.8545 \\
&= 24.1045
\end{align*}
\][/tex]
So, the value of [tex]\( y \)[/tex] for [tex]\( x = 4 \)[/tex] is approximately [tex]\( 24 \)[/tex].
Hence, the correct answer is:
C. 24
