A satellite dish has a parabolic cross-section. The dish is 12 feet wide at the opening and 2 feet deep. Find the equation of the parabola if the vertex is at the origin.

Given that the satellite dish has a parabolic cross-section, and we know the width at the opening and the depth, we can determine the equation of the parabola. 

### Step-by-step solution:

1. **Parabola equation:**  
   Since the vertex is at the origin \((0,0)\) and the parabola opens upwards, the standard form of the equation is:
   \[
   y = ax^2
   \]
   where \(a\) is a constant to be determined.

2. **Key points from the problem:**  
   - The dish is **12 feet wide** at the opening, so the distance from the center to the edge is \(6\) feet (half of 12 feet). This gives us the point \((6, 2)\), since at \(x = 6\), the depth \(y = 2\) feet.
   - The depth is \(2\) feet, meaning when \(x = 6\), \(y = 2\).

3. **Substitute the point \((6, 2)\) into the equation:**
   \[
   2 = a(6^2)
   \]
   \[
   2 = 36a
   \]
   Solving for \(a\):
   \[
   a = \frac{2}{36} = \frac{1}{18}
   \]

4. **Final equation:**
   Therefore, the equation of the parabola is:
   \[
   y = \frac{1}{18}x^2
   \]

This is the equation representing the parabolic shape of the satellite dish.

Would you like more details or have any questions?

Here are 5 related questions to explore further:
1. How would the equation change if the dish was 15 feet wide instead of 12?
2. Can you find the focus of the parabola given this equation?
3. How would the depth of the dish change if the equation was \(y = \frac{1}{25}x^2\)?
4. What is the significance of the parameter \(a\) in the parabola’s equation?
5. Can you derive the focal length of the parabola from this equation?

**Tip:** When solving for the equation of a parabola, it's useful to know key points on the parabola such as the vertex and another point through which the parabola passes.

Discover how to find the equation of a parabola for a satellite dish that is 12 feet wide and 2 feet deep, with the vertex at the origin. This problem involves key geometric and algebraic concepts, including solving quadratic equations. Suitable for high school students in Grades 9-12.

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