This problem involves determining the heights of posts placed at intervals along a parabolic arc supporting a bridge. Here's a breakdown of the situation:

• The highest posts are 10 meters tall.
• The parabolic arc has its vertex at 6 meters above the base level of the bridge (so the total height of the arc at the vertex is 16 meters, as 10 meters of posts plus the 6 meters from the bridge level).
• The distance between the ends of the bridge, where the parabola touches the ground, is 60 meters.

Let's set up the equation of the parabola:

Step 1: Define a coordinate system

• Place the origin of the coordinate system at the vertex of the parabola, which is at $(0, 16)$.
• The ends of the parabola are 30 meters away from the vertex on either side (because the total distance is 60 meters).
• The general form of a parabola is: $y = ax^2 + bx + c$ Since the vertex is at $(0, 16)$, we have: $y = a(x - 0)^2 + 16 \quad \Rightarrow \quad y = ax^2 + 16$

Step 2: Use the boundary condition

• At the ends of the bridge (30 meters away from the vertex), the height of the parabola is 6 meters (since the posts are resting on the arc, and the tip of the arc is 6 meters above the bridge).
• This gives the condition: $y(30) = 6$ Substituting into the equation: $6 = a(30)^2 + 16$ Solving for $a$: $6 = 900a + 16$ $900a = -10 \quad \Rightarrow \quad a = -\frac{10}{900} = -\frac{1}{90}$ Therefore, the equation of the parabola is: $y = -\frac{1}{90}x^2 + 16$

Step 3: Calculate the height of posts at intervals

Now that we have the equation of the parabola, we can calculate the height of the posts every 10 meters along the bridge. The positions of the posts will be at $x = -30, -20, -10, 0, 10, 20, 30$ meters from the vertex.

• For $x = -30$: $y(-30) = -\frac{1}{90}(-30)^2 + 16 = 6$
• For $x = -20$: $y(-20) = -\frac{1}{90}(-20)^2 + 16 = -\frac{1}{90}(400) + 16 = 11.56 \text{ meters}$
• For $x = -10$: $y(-10) = -\frac{1}{90}(-10)^2 + 16 = -\frac{1}{90}(100) + 16 = 14.89 \text{ meters}$
• For $x = 0$ (at the vertex): $y(0) = 16 \text{ meters}$
• For $x = 10$: $y(10) = -\frac{1}{90}(10)^2 + 16 = 14.89 \text{ meters}$
• For $x = 20$: $y(20) = -\frac{1}{90}(20)^2 + 16 = 11.56 \text{ meters}$
• For $x = 30$: $y(30) = 6 \text{ meters}$

Final Result:

The heights of the posts every 10 meters along the bridge, starting from the ends (where the arc is 6 meters above the bridge) to the center (vertex), are:

• 6 meters
• 11.56 meters
• 14.89 meters
• 16 meters (at the vertex)
• 14.89 meters
• 11.56 meters
• 6 meters

Would you like a more detailed explanation or clarification?

Here are five related questions that could extend your understanding:

1. What is the general form of a parabolic equation, and how is it derived from vertex information?
2. How does changing the distance between the posts affect the calculation of post heights?
3. How would the equation of the parabola change if the arc were higher or lower?
4. Could you use a different coordinate system or method to find the height of the posts?
5. How would this scenario change if the arc were modeled as a different curve, like a sine wave?

Tip: When solving for a quadratic function from a real-world context like this, always ensure to understand where the vertex and boundary conditions are applied to set up the correct equation.