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The function given in the problem is \( f(x, y) = \sin \left(\frac{2x^2 - y^2}{10}\right) \). You need to find which level curve corresponds to \( c = \frac{1}{2} \). This means you're solving for:

\[
\sin \left(\frac{2x^2 - y^2}{10}\right) = \frac{1}{2}
\]

We know from trigonometry that \( \sin(\pi/6) = \frac{1}{2} \), so we look for angles where the sine value is \( \frac{1}{2} \), which would be \( \pi/6, 5\pi/6, \ldots \). This translates to finding the argument of sine that matches these values, multiplied by \( 10 \) to cancel the denominator. So, the next step is to solve for:

\[
\frac{2x^2 - y^2}{10} = \frac{\pi}{6} + k\pi \text{ for integers } k
\]

The simplest case occurs when \( k = 0 \):
\[
2x^2 - y^2 = \frac{10\pi}{6} = \frac{5\pi}{3}
\]

Now, we need to match this to one of the provided options (which I assume are to be exact matches for simplicity):

A) \( x^2 - \frac{y^2}{2} = 2.618 \) converts to \( 2x^2 - y^2 = 5.236 \) (which is not \( \frac{5\pi}{3} \))

B) \( 2x^2 - y^2 = 5 \) (which is clearly not \( \frac{5\pi}{3} \))

C) \( -2x^2 + y^2 = \sqrt{5} \) (rearranging signs and values, does not match)

D) \( x^2 + 2y^2 = 4 \) converts to \( 2x^2 + 4y^2 = 8 \) (which again, does not match)

Since none of these options exactly equal \( \frac{5\pi}{3} \approx 5.236 \), it's possible there might be an error in the transcription or interpretation of the options. It's also important to consider rounding errors or approximations that might have been applied to the options, such as in option A, which is the closest.

Would you like a more detailed explanation of any steps here or have any other questions on this topic? Here are a few further questions to consider:

1. Would you like to know how to derive the general form of a level curve for a trigonometric function like this one?
2. How can you apply this knowledge to other trigonometric functions and their level curves?
3. Are you interested in learning about the implications of choosing different values for \( k \) in the level curve equation?
4. Would you like a detailed review on how to manipulate equations to match a desired form?
5. How can sine values other than \( \frac{1}{2} \) affect the shape and equation of level curves?

**Tip**: Always remember that the argument of the sine function must be handled carefully, especially when working with periodic functions where multiple angles can yield the same sine value.

¿Cuál de las siguientes curvas de nivel corresponde a la curva de nivel con c = 1/2 de la función de dos variables f(x, y) = sin((2x^2 - y^2) / 10)?

This problem focuses on determining the level curve for c = 1/2 of the function f(x, y) = sin((2x^2 - y^2) / 10). Using trigonometric identities and algebraic manipulation, solve for the corresponding curve. Suitable for college-level mathematics.

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