To analyze the relationship between amplitude and energy and find the missing data point for amplitude 9 units, we follow these steps:
### Step 1: Understanding the Relationship
From the given data:
[tex]\[
\begin{array}{|c|c|}
\hline
\text{Amplitude} & \text{Energy} \\
\hline
6 & 72 \\
\hline
7 & 98 \\
\hline
8 & 128 \\
\hline
9 & ? \\
\hline
10 & 200 \\
\hline
\end{array}
\][/tex]
We assume a quadratic relationship between amplitude ([tex]\(A\)[/tex]) and energy ([tex]\(E\)[/tex]). That is, [tex]\(E = k \cdot A^2\)[/tex], where [tex]\(k\)[/tex] is a proportionality constant.
### Step 2: Squaring the Amplitudes
Calculate the square of each amplitude value as follows:
[tex]\[
\begin{array}{|c|c|c|}
\hline
\text{Amplitude} & \text{Energy} & \text{Amplitude}^2 \\
\hline
6 & 72 & 36 \\
\hline
7 & 98 & 49 \\
\hline
8 & 128 & 64 \\
\hline
9 & ? & 81 \\
\hline
10 & 200 & 100 \\
\hline
\end{array}
\][/tex]
### Step 3: Performing Linear Regression
By regressing the squared amplitudes against the energy values, we identify the linear relationship. This can be expressed as:
[tex]\[
E = m \cdot (\text{Amplitude}^2) + c
\][/tex]
where [tex]\(m\)[/tex] and [tex]\(c\)[/tex] are the coefficients of the regression.
### Step 4: Determining the Coefficients
The regression yields the following coefficients:
[tex]\[
m = 2.0, \quad c \approx -5.899 \cdot 10^{-14}
\][/tex]
So the equation is:
[tex]\[
E = 2 \cdot (\text{Amplitude}^2) - 5.899 \cdot 10^{-14}
\][/tex]
### Step 5: Predicting the Missing Energy Value
For an amplitude of 9 units, we calculate the energy:
[tex]\[
\text{Amplitude}^2 = 9^2 = 81
\][/tex]
Substitute 81 into the regression equation:
[tex]\[
E = 2 \cdot 81 - 5.899 \cdot 10^{-14}
\][/tex]
[tex]\[
E = 162 - 5.899 \cdot 10^{-14} \approx 162
\][/tex]
### Conclusion
The energy corresponding to an amplitude of 9 units is approximately 162 units. Hence, the complete data table is:
[tex]\[
\begin{array}{|c|c|}
\hline
\text{Amplitude} & \text{Energy} \\
\hline
6 & 72 \\
\hline
7 & 98 \\
\hline
8 & 128 \\
\hline
9 & 162 \\
\hline
10 & 200 \\
\hline
\end{array}
\][/tex]
