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This problem involves a vehicle (a heavy transporter) trying to pass under a parabolic arch bridge. We need to determine if the vehicle, which is 10.0 meters wide and 4.8 meters tall, can fit under the bridge. The bridge spans 24.0 meters with a maximum height of 6.0 meters at the center. The goal is to first establish the equation of the parabolic arch.

**Step 1: Setup the Coordinate System and Equation**

We can define a coordinate system where the vertex of the parabola (the highest point of the arch) is at the origin. This simplifies the vertex form of the parabola, which can be expressed as:
\[ y = a(x - h)^2 + k \]
For the vertex \( (h, k) \), \( h = 0 \) and \( k = 6.0 \) meters (since the bridge is highest at its center, and we're placing this point at the origin).

Now, let's place the x-axis along the bottom of the bridge span. Since the total span is 24 meters, the endpoints of the parabola on the x-axis will be at \( x = -12 \) and \( x = 12 \) (assuming the center at \( x = 0 \) and symmetry about this point).

**Step 2: Determine the Constant \( a \)**

The general form of the vertex equation adjusted to our coordinate system is:
\[ y = a(x - 0)^2 + 6 \]
\[ y = ax^2 + 6 \]

Given that the bridge meets the ground (or base of the parabolic arch) at \( x = 12 \) meters and \( y = 0 \) (ground level),
\[ 0 = a(12)^2 + 6 \]
\[ a(144) + 6 = 0 \]
\[ a = -\frac{6}{144} = -\frac{1}{24} \]

Thus, the equation of the parabola becomes:
\[ y = -\frac{1}{24}x^2 + 6 \]

**Step 3: Checking Clearance**

With the parabolic equation known, we can check if the vehicle, which is 4.8 meters tall, fits under the bridge at its full width of 10 meters. Given the vehicle's width, we need to evaluate the parabolic height at \( x = \pm 5 \) meters (half the vehicle width to either side of the center):
\[ y = -\frac{1}{24}(5)^2 + 6 \]
\[ y = -\frac{1}{24}(25) + 6 \]
\[ y = -\frac{25}{24} + 6 \]
\[ y = 6 - \frac{25}{24} \]
\[ y = 6 - 1.0417 \]
\[ y \approx 4.9583 \text{ meters} \]

**Conclusion:**
The calculated height of the arch at 5 meters from the center is approximately 4.9583 meters, which is just slightly higher than the vehicle's height of 4.8 meters. Therefore, the vehicle should be able to pass under the bridge.

Would you like to delve into any other details or have any other questions? Here are a few topics we might explore:

1. How would the height at different points along the bridge width affect larger or differently shaped vehicles?
2. How does changing the shape or dimensions of the arch affect the equation of the parabola?
3. What are the structural considerations for building a bridge with such a parabolic arch?
4. Can we explore the real-world applications of parabolic arches beyond bridge design?
5. How would the structural integrity of the bridge be affected if the maximum height was different?

**Tip:** When solving real-world problems using mathematical models like parabolas, always consider adding a safety margin to your calculations to account for potential inaccuracies and ensure safety.

Ein Schwertransporter ist 10,0 m breit und 4,8 m hoch. Kann er unter der Brücke durchfahren? Ermittle zunächst die Funktionsgleichung für den parabelförmigen Brückenbogen.

Determine whether a 10-meter-wide and 4.8-meter-tall transporter can pass under a parabolic arch bridge with a span of 24 meters and a maximum height of 6 meters. Learn how to derive the parabolic equation and calculate the clearance for the vehicle.

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