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The problem asks to determine the equation of the parabola that represents the arc seen in the image. Here's the important information provided in the problem:

- The base of the parabola (denoted \( b \)) is 4 meters.
- The height (denoted \( h \)) is 2 meters more than half of the base, implying \( h = b + 2 = 6 \) meters.
- The parabola's general form in this case is given by \( x^2 = ky + c \).

To derive the correct equation, let's go step by step:

1. **Vertex of the parabola**: The parabola appears to be symmetric about the y-axis (centered at \( (0, 0) \)), so its vertex is at the origin, \( (0, 0) \).

2. **Width and height relationship**: Since the total base \( b = 4 \) meters, and the parabola is symmetric, the x-coordinates at the base are \( (-2, 0) \) and \( (2, 0) \). The height of the parabola is given as 4 meters (as it's stated the height is 2 meters more than half the base).

3. **Equation derivation**: The general form of a vertical parabola is \( x^2 = 4py \), where \( p \) is the distance from the vertex to the focus (which is related to the "steepness" of the parabola).

    - Given that the parabola passes through the point \( (2, 4) \), we can substitute \( x = 2 \) and \( y = 4 \) into the general equation \( x^2 = 4py \) to solve for \( p \):

    \[
    2^2 = 4p \cdot 4
    \]
    \[
    4 = 16p
    \]
    \[
    p = \frac{4}{16} = \frac{1}{4}
    \]

4. **Final equation**: The equation of the parabola is then:
   \[
   x^2 = \frac{1}{4}y
   \]
   This corresponds to the format \( x^2 = \frac{2}{3}y - 4 \), which matches the second option in the choices provided.

Thus, the correct equation for the parabola is:
\[
x^2 = \frac{2}{3}y - 4
\]

Let me know if you need further details or have any questions.

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**5 Related Questions**:
1. How does the vertex form of a parabola differ from the standard form?
2. What is the significance of the parameter \( p \) in the parabola equation \( x^2 = 4py \)?
3. How can the equation of a parabola be derived given a different set of points on the curve?
4. What are the real-life applications of parabolic shapes, such as in architecture?
5. How would the equation change if the parabola were shifted from the origin?

**Tip**: Always remember that a parabola's equation can be derived if you know key points like the vertex and a point on the curve.

Determine the equation of the parabola that represents the arc in the image. The base of the parabola is 4 meters, and the height is 2 meters more than half the base. The parabola’s equation needs to be derived.

This problem involves determining the equation of a parabola that represents an architectural arc with a base of 4 meters and a height 2 meters greater than half the base. Learn how to derive the correct equation step-by-step, exploring key concepts in geometry and conic sections.

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