To analyze the mathematical relationship between the amplitude and energy of a mechanical wave, let's start by examining the data provided in the table:
\begin{tabular}{|l|l|}
\hline
Amplitude (in units) & Energy (in units) \\
\hline
1 & 2 \\
\hline
2 & 8 \\
\hline
3 & 18 \\
\hline
4 & 32 \\
\hline
5 & 50 \\
\hline
\end{tabular}
From this data, we observe that a mechanical wave at an amplitude of 1 unit has an energy of 2 units, and a mechanical wave at an amplitude of 2 units has an energy of 8 units. To find the relationship between the energy at these two amplitudes, we will compare their energy values.
First, let's restate the given data:
- Mechanical wave [tex]\(A\)[/tex] with an amplitude of 1 cm has an energy of 2 units.
- Mechanical wave [tex]\(B\)[/tex] with an amplitude of 2 cm has an energy of 8 units.
To find the relationship between the energy of wave [tex]\(A\)[/tex] and wave [tex]\(B\)[/tex], we calculate the ratio of their energies:
[tex]\[
\text{Relationship} = \frac{\text{Energy of wave } B}{\text{Energy of wave } A}
\][/tex]
Substituting the corresponding energy values:
[tex]\[
\text{Relationship} = \frac{8 \text{ units}}{2 \text{ units}} = 4
\][/tex]
This result indicates that the energy carried by mechanical wave [tex]\(B\)[/tex], with an amplitude of 2 cm, is 4 times the energy carried by mechanical wave [tex]\(A\)[/tex], with an amplitude of 1 cm.
Thus, the correct answer is:
C. The amount of energy in wave [tex]\(B\)[/tex] is four times the amount of energy in wave [tex]\(A\)[/tex].
