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A lab team was collecting data on the amplitude and energy of a mechanical wave, as shown in the data table. They forgot to record one data point.

Analyze the data to identify the mathematical relationship between amplitude and energy. Calculate the missing data point.

\begin{tabular}{|l|l|}
\hline Amplitude & Energy \\
\hline 6 units & 72 units \\
\hline 7 units & 98 units \\
\hline 8 units & 128 units \\
\hline 9 units & [tex]$?$[/tex] units \\
\hline 10 units & 200 units \\
\hline
\end{tabular}

Sagot :

GinnyAnsweridn
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]
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