To identify the mathematical relationship between the amplitude and energy of a mechanical wave, let's carefully analyze the given data and determine the pattern that relates amplitude to energy.
Given data:
[tex]\[
\begin{array}{|l|l|}
\hline
\text{Amplitude (units)} & \text{Energy (units)} \\
\hline
1 & 2 \\
\hline
2 & 8 \\
\hline
3 & 18 \\
\hline
4 & 32 \\
\hline
5 & 50 \\
\hline
\end{array}
\][/tex]
Let's start by looking at the amplitudes and corresponding energies for two specific mechanical waves:
- Mechanical wave [tex]\(A\)[/tex] with amplitude [tex]\(1\)[/tex] unit has energy [tex]\(2\)[/tex] units.
- Mechanical wave [tex]\(B\)[/tex] with amplitude [tex]\(2\)[/tex] units has energy [tex]\(8\)[/tex] units.
To determine the relationship between the energy carried by these two waves, we must find the ratio of the energy of wave [tex]\(B\)[/tex] to the energy of wave [tex]\(A\)[/tex]:
1. Identify the energy of wave [tex]\(A\)[/tex]:
[tex]\[
E_A = 2\ \text{units}
\][/tex]
2. Identify the energy of wave [tex]\(B\)[/tex]:
[tex]\[
E_B = 8\ \text{units}
\][/tex]
3. Calculate the ratio of [tex]\(E_B\)[/tex] to [tex]\(E_A\)[/tex]:
[tex]\[
\frac{E_B}{E_A} = \frac{8}{2} = 4.0
\][/tex]
This ratio [tex]\(4.0\)[/tex] indicates that the energy in wave [tex]\(B\)[/tex] is four times the energy in wave [tex]\(A\)[/tex].
Therefore, the correct relationship is:
[tex]\[ \text{The amount of energy in wave } B \text{ is four times the amount of energy in wave } A. \][/tex]
Thus, the correct answer is:
[tex]\[ \mathbf{A. \text{The amount of energy in wave B is four times the amount of energy in wave A.}} \][/tex]
