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The image contains a mathematical word problem in German related to geometry and construction.

Here’s a translation of the problem:

**Problem:**
"A cable bridge is to be constructed. The span (distance between the two main pillars) of the bridge is 240 meters. The height of the two main pillars above the bridge deck is 30 meters, and the cables have a sag of 50 meters. Determine the functional equation for the cable's suspension curve and calculate the length of the cable."

In this case, it appears the problem is asking to model the cable suspension using a parabola equation, which is typical in such bridge problems. The information provided allows us to determine the dimensions of the parabola and its vertex form.

### Approach:
1. **Equation Setup**: Since this is a typical suspension bridge problem, the shape of the cable between the pillars can be modeled as a parabola. We will assume that the lowest point of the cable is at the midpoint of the span (the sag point).
2. **Define Coordinate System**: Place the origin of a Cartesian plane at the lowest point of the cable (the middle of the bridge) so that the parabola opens upwards. 
   - The vertex of the parabola will be at \( (0, -50) \), since the sag is 50 meters below the deck.
   - The two points at the top of the pillars are located at \( (-120, 30) \) and \( (120, 30) \), since the total span is 240 meters.

### Parabola Equation:
The standard form of a parabola is given by:
\[ y = a(x - h)^2 + k \]
where \( (h, k) \) is the vertex of the parabola. Here, the vertex is \( (0, -50) \), so the equation simplifies to:
\[ y = a x^2 - 50 \]

Next, use the known point at \( (120, 30) \) to solve for \( a \):
\[
30 = a(120)^2 - 50
\]
\[
30 + 50 = a(14400)
\]
\[
80 = 14400a
\]
\[
a = \frac{80}{14400} = \frac{1}{180}
\]

Thus, the equation of the parabola is:
\[
y = \frac{1}{180}x^2 - 50
\]

### Cable Length:
To find the length of the cable, we calculate the arc length of the parabola from \( x = -120 \) to \( x = 120 \). The arc length \( L \) of a curve \( y = f(x) \) between \( x = a \) and \( x = b \) is given by:
\[
L = \int_a^b \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx
\]

Let me now calculate this arc length for the given function.The length of the cable is approximately 298.88 meters.

This gives the total length of the cable between the two main pillars of the bridge.

Would you like further details or have any other questions? Here are some related questions you might find interesting:

1. How can the arc length formula be derived?
2. What other geometric curves are used in bridge construction?
3. Can we calculate the tension in the cable based on this model?
4. How does the sag affect the overall stability of the bridge?
5. How would the parabola change if the height of the pillars or sag were different?

**Tip**: Suspension cables often follow a parabolic curve, while the distribution of forces along the cable follows a catenary curve, which is slightly different!

A cable bridge is to be constructed. The span (distance between the two main pillars) of the bridge is 240 meters. The height of the two main pillars above the bridge deck is 30 meters, and the cables have a sag of 50 meters. Determine the functional equation for the cable's suspension curve and calculate the length of the cable.

Solve the cable suspension bridge problem by determining the parabola equation based on the given span and sag values, and calculate the length of the cable using integral arc length formulas. Suitable for high school students in Grades 11-12.

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