a group of engineers are building a paraboolic satellite dish whose shape will be formed by rotating the curve around the yxais if the dish has a 8 foot diamter and a max depth of 2 feet find the value of a if the curve is y=ax^2

To find the value of \( a \) in the equation \( y = ax^2 \), given the satellite dish has a diameter of 8 feet and a maximum depth of 2 feet, we can follow these steps:

### Step 1: Interpret the problem.
- The diameter of the dish is 8 feet, meaning the radius \( r = \frac{8}{2} = 4 \) feet.
- The maximum depth is 2 feet, which represents the \( y \)-value when \( x = 4 \).

### Step 2: Apply the known values.
At the edge of the dish, when \( x = 4 \), the depth \( y = 2 \).

Using the equation \( y = ax^2 \):
\[
y = a(4)^2 = 2
\]

This simplifies to:
\[
16a = 2
\]

### Step 3: Solve for \( a \).
\[
a = \frac{2}{16} = \frac{1}{8}
\]

Thus, the value of \( a \) is \( \frac{1}{8} \).

### Conclusion:
The equation of the parabola forming the dish is \( y = \frac{1}{8}x^2 \).

Would you like further details or have any questions?

Here are some related questions to explore this topic further:
1. What is the volume of the satellite dish formed by rotating this parabola around the y-axis?
2. How would the equation change if the maximum depth were doubled?
3. How can you use the focal point of the parabola to improve signal reflection in a satellite dish?
4. How would the value of \( a \) change if the diameter were reduced to 4 feet but the depth remained the same?
5. Can you derive a general formula for \( a \) in terms of the diameter and depth?

**Tip:** When solving problems involving curves, always consider the geometric properties, such as symmetry and dimensions, to set up equations effectively.

A group of engineers are building a parabolic satellite dish whose shape will be formed by rotating the curve around the y-axis. If the dish has an 8-foot diameter and a maximum depth of 2 feet, find the value of a if the curve is y = ax^2.

Learn how to find the value of 'a' for a parabolic satellite dish with an 8-foot diameter and a 2-foot maximum depth. This problem explores algebraic concepts, such as quadratic equations and properties of parabolas, and applies them to a real-world engineering scenario. Suitable for students in Grades 10-12.

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