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