To determine the relationship between the amplitude of the mechanical wave and the energy it carries, let's analyze the data from the given table:
[tex]\[
\begin{array}{|c|c|}
\hline
\text{Amplitude (units)} & \text{Energy (units)} \\
\hline
1 & 2 \\
\hline
2 & 8 \\
\hline
3 & 18 \\
\hline
4 & 32 \\
\hline
\end{array}
\][/tex]
By closely examining the data, we are looking for a possible consistent mathematical relationship between the amplitude ([tex]\( A \)[/tex]) and the energy ([tex]\( E \)[/tex]). One approach to find the relationship is to try different mathematical models, starting with the simplest ones. In this case, since energy seems to increase with the square of the amplitude, we assume a quadratic relationship of the form:
[tex]\[ E = k \times A^2 \][/tex]
where [tex]\( k \)[/tex] is a constant.
We can determine the constant [tex]\( k \)[/tex] using one of the given data points. Let's use the first data point where [tex]\( A = 1 \)[/tex] unit and [tex]\( E = 2 \)[/tex] units:
[tex]\[ 2 = k \times 1^2 \][/tex]
[tex]\[ k = 2 \][/tex]
Let's verify this constant with other data points:
- For [tex]\( A = 2 \)[/tex] units:
[tex]\[ 8 = k \times 2^2 \][/tex]
[tex]\[ 8 = k \times 4 \][/tex]
[tex]\[ k = 2 \][/tex]
- For [tex]\( A = 3 \)[/tex] units:
[tex]\[ 18 = k \times 3^2 \][/tex]
[tex]\[ 18 = k \times 9 \][/tex]
[tex]\[ k = 2 \][/tex]
- For [tex]\( A = 4 \)[/tex] units:
[tex]\[ 32 = k \times 4^2 \][/tex]
[tex]\[ 32 = k \times 16 \][/tex]
[tex]\[ k = 2 \][/tex]
The constant [tex]\( k = 2 \)[/tex] consistently holds true for all given data points. Thus, the relationship can be established as:
[tex]\[ E = 2 \times A^2 \][/tex]
Now, let's use this equation to find the energy if the amplitude is [tex]\( 9 \)[/tex] units:
[tex]\[ E = 2 \times 9^2 \][/tex]
[tex]\[ E = 2 \times 81 \][/tex]
[tex]\[ E = 162 \text{ units} \][/tex]
So, the energy corresponding to an amplitude of 9 units is 162 units.
Among the given choices:
A. 324 units
B. 54 units
C. 98 units
D. 162 units
The correct answer is:
D. 162 units
