The towers of a suspension bridge are 800 feet apart and rise 160 feet above the road. The cable between the towers has the shape of a parabola and the cable just touches the sides of the road midway between the towers. What is the height of the cable 100 feet from a tower?

To find the height of the cable 100 feet from a tower, we need to use the equation of a parabola. Let's first establish the key information and set up the coordinate system.

### Problem Setup:
1. The distance between the towers is 800 feet.
2. The towers rise 160 feet above the road.
3. The cable just touches the road midway between the towers.
   
We can set up the coordinate system with:
- The road as the \( x \)-axis.
- The point midway between the towers as the origin \( (0, 0) \) of the coordinate system.
- The two towers at \( x = -400 \) feet and \( x = 400 \) feet.

Since the cable is parabolic and touches the road at the midpoint, the equation of the parabola can be written in the form:
\[
y = a x^2 + c
\]
where \( y \) is the height of the cable, \( x \) is the horizontal distance from the midpoint, \( a \) is a constant to be determined, and \( c \) is the height of the cable at the midpoint (which is 0, since the cable just touches the road at \( x = 0 \)).

Thus, the equation simplifies to:
\[
y = a x^2
\]

### Finding the value of \( a \):
At \( x = 400 \) feet (the distance from the midpoint to the tower), the height of the cable is 160 feet. Plugging these values into the equation:

\[
160 = a (400)^2
\]
\[
160 = a \times 160000
\]
\[
a = \frac{160}{160000} = \frac{1}{1000}
\]

So, the equation of the cable is:
\[
y = \frac{1}{1000} x^2
\]

### Finding the height 100 feet from a tower:
Now, we need to find the height of the cable 100 feet from a tower. Since the tower is at \( x = 400 \), 100 feet from the tower corresponds to \( x = 400 - 100 = 300 \) feet.

Substitute \( x = 300 \) into the equation:
\[
y = \frac{1}{1000} (300)^2
\]
\[
y = \frac{1}{1000} \times 90000 = 90
\]

### Conclusion:
The height of the cable 100 feet from a tower is 90 feet.

---

Would you like further details or have any questions about the process?

Here are five related questions to deepen your understanding:
1. How would the equation change if the cable did not touch the road but was suspended a certain distance above it?
2. What would the height of the cable be 200 feet from the midpoint of the bridge?
3. Can you generalize the equation for different distances between towers and tower heights?
4. How would the problem change if the cable formed a catenary instead of a parabola?
5. Can you calculate the length of the cable between the two towers?

**Tip**: For problems involving parabolas, always try to first identify the vertex and the general form of the equation before plugging in the known points to find the constant terms.

Explore how to find the height of a suspension bridge's cable, shaped as a parabola, 100 feet from a tower. This problem involves algebra, quadratic equations, and the properties of parabolas, making it suitable for students in Grades 10-12. Learn to set up and solve the equation of a parabola with real-world applications.

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