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The problem describes a suspension bridge model with two towers and a parabolic cable. Let's break down the problem:

### Problem Breakdown:
1. **Towers and Positions:**
   - The towers are 20 meters apart, each positioned 10 meters away from the origin.
   - Each tower is 12 meters high.

2. **Cable:**
   - The cable forms a parabola between the two towers.
   - The lowest point of the cable is 3 meters above the origin.
   - The problem asks for the vertical distance of the cable 5 meters away from the center.

### Solution Approach:
Since the cable is parabolic, we can represent its shape using the standard form of a parabola:

\[
y = ax^2 + bx + c
\]

Given conditions:
- At the origin (center of the bridge), \( x = 0 \), the lowest point \( y = 3 \). This gives \( c = 3 \).
- At the towers (10 meters away), the cable is at the top height of 12 meters, i.e., when \( x = \pm 10 \), \( y = 12 \).

Using these, we solve for \( a \) and \( b \).

### Finding \(a\) and \(b\):
1. At \( x = 10 \):
   
   \[
   12 = a(10)^2 + b(10) + 3
   \]
   
   Simplifying:
   
   \[
   12 = 100a + 10b + 3
   \]

   Which simplifies to:

   \[
   100a + 10b = 9 \quad \text{(Equation 1)}
   \]

2. At \( x = -10 \):

   \[
   12 = a(-10)^2 + b(-10) + 3
   \]
   
   Simplifying:

   \[
   12 = 100a - 10b + 3
   \]

   Which simplifies to:

   \[
   100a - 10b = 9 \quad \text{(Equation 2)}
   \]

Adding Equation 1 and Equation 2:

\[
200a = 18 \implies a = 0.09
\]

Substituting \( a = 0.09 \) into Equation 1:

\[
100(0.09) + 10b = 9 \implies 9 + 10b = 9 \implies b = 0
\]

Thus, the equation of the parabola is:

\[
y = 0.09x^2 + 3
\]

### Finding the Vertical Distance at \( x = 5 \):
Substitute \( x = 5 \):

\[
y = 0.09(5)^2 + 3 = 0.09(25) + 3 = 2.25 + 3 = 5.25
\]

Therefore, the vertical distance of the cable 5 meters away from the center is **5.25 meters**.

---

Do you need further explanation or details on any step? Here are some related questions:

1. How can you determine the parabola's vertex from its equation?
2. What would the equation look like if the towers were positioned differently?
3. How does changing the height of the towers affect the parabola?
4. What other forms can be used to represent parabolas in bridge designs?
5. Can we calculate the horizontal tension in the cable using this model?

**Tip:** When working with parabolas in physical contexts, always identify key points like the vertex and use known values to find coefficients systematically.

What is the vertical distance of the cable 5 meters away from the center of a suspension bridge model with towers 20 meters apart and 12 meters high, where the cable is parabolic with its lowest point 3 meters above the origin?

Explore how to determine the vertical distance of a cable 5 meters away from the center in a suspension bridge model. This problem involves quadratic functions and parabolas, requiring knowledge of algebra and coordinate geometry, suitable for students in Grades 10-12.

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