To determine the relationship between amplitude and energy, we will analyze the provided data. The table shows measurements of amplitude and corresponding energy values:
[tex]\[
\begin{array}{|c|c|}
\hline
\text{Amplitude (A)} & \text{Energy (E)} \\
\hline
1 & 2 \\
\hline
2 & 8 \\
\hline
3 & 18 \\
\hline
4 & 32 \\
\hline
\end{array}
\][/tex]
We will fit these data points to a polynomial of the form [tex]\( E = a \cdot A^2 + b \cdot A + c \)[/tex]. Solving the regression for the quadratic polynomial, we find the coefficients [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex]. In our case, this relationship simplifies to primarily a quadratic dependence:
[tex]\[ E \approx 2 \cdot A^2 \][/tex]
Given this relationship, we'll now use it to find the energy when the amplitude is 9 units:
[tex]\[ E = 2 \cdot A^2 \][/tex]
Substituting [tex]\( A = 9 \)[/tex]:
[tex]\[ E = 2 \cdot (9)^2 \][/tex]
[tex]\[ E = 2 \cdot 81 \][/tex]
[tex]\[ E = 162 \][/tex]
Therefore, if the amplitude is 9 units, the energy is 162 units.
The correct answer is:
D. 162 units
