The data given shows the amplitude and energy of several simple waves. Information,begin bold,Amplitude of,end bold,,begin bold,wave ,end bold,(centimeters),begin bold,Energy of,end bold,,begin bold,wave,end bold, (Joules)0.51.01.04.01.59.0 Question Based on the data, what should be the amount of energy for a simple wave with an amplitude of 3.0 centimeters?Answer options with 4 options.12.0 Joules.16.0 Joules.18.0 Joules.36.0 Joules

1 answer

To determine the energy of a simple wave based on the given amplitude and energy data, we can look for a pattern or relationship between amplitude and energy.

From the data provided:

  • Amplitude 0.5 cm → Energy 1.0 J
  • Amplitude 1.0 cm → Energy 4.0 J
  • Amplitude 1.5 cm → Energy 9.0 J

Notice that the energy seems to be proportional to the square of the amplitude.

Let's summarize the relationship we can observe:

  • For amplitude 0.5 cm: \( E = 1 \times (0.5)^2 = 1.0 \) J
  • For amplitude 1.0 cm: \( E = 4 \times (1.0)^2 = 4.0 \) J
  • For amplitude 1.5 cm: \( E = 9 \times (1.5)^2 = 9.0 \) J

This suggests that the energy \( E \) might be calculated using the equation \( E = k \times A^2 \), where \( k \) is a constant and \( A \) is the amplitude. From the previous observation, we can conclude that \( E \) is proportional to the square of the amplitude.

We need to find the energy corresponding to an amplitude of 3.0 cm:

  1. Calculate \( 3.0^2 = 9.0 \).

  2. Based on the previous data, we can see that when amplitude increases from 1.5 cm (9.0 J) to 3.0 cm, we can estimate:

    • If we consider a simple linear relation, it seems like \( E \) increases quadratically (allows for scaling).

Since \( 3.0^2 = 9 \) corresponds to squared amplitude, you can scale the energy values accordingly based on our previous observations.

The energies appear to scale:

  • \( 0.5^2 : 1.0 \) J
  • \( 1.0^2 : 4.0 \) J
  • \( 1.5^2 : 9.0 \) J
  • \( 3.0^2 : E \)

Where we can estimate if the pattern holds, \( E(3.0) \) should relate similarly. The scaling might inform that:

  • Doubling the amplitude from 1.5 cm (9.0 J) to 3.0 cm will yield:

\[ E(3) = 4 \times 2.25 \text{ (being } \frac{9}{4})= 9 \times 4 = 36 \text{ J} \]

Thus, 36.0 Joules is a reasonable corresponding energy level where we see a stability of the amplitude against the pattern.

Based on this, the correct answer is: 36.0 Joules.

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