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This problem, called "Das Bobbahn Problem," involves modeling a bobsled track using a quadratic function. To develop the function that describes the track, three conditions are needed. Two conditions are already provided:

1. The height of the starting point is \( (0, 20) \).
2. The lowest point is located at \( (30, 0) \).

The third condition comes from the slope derived from the angle, \( 40^\circ \), near the top of the ramp, which will help find the slope of the curve at \( x = 0 \).

### Steps for solving the problem:

1. **Determine the slope from the angle:**
   - The slope \( m \) at the top of the ramp can be found using the tangent of the angle \( 40^\circ \).
   \[
   m = \tan(40^\circ)
   \]
2. **Model the function as a quadratic equation:**
   - A general quadratic function is \( f(x) = ax^2 + bx + c \).
   - Using the known points and slope, substitute them into this equation to find the constants \( a \), \( b \), and \( c \).

3. **Substitute the known points and slope:**
   - For the point \( (0, 20) \), substitute \( x = 0 \) and \( f(x) = 20 \) into the quadratic equation.
   - For the point \( (30, 0) \), substitute \( x = 30 \) and \( f(x) = 0 \).
   - Use the slope at \( x = 0 \) (calculated from \( \tan(40^\circ) \)) to derive an equation involving \( a \) and \( b \).

Would you like a more detailed breakdown of the mathematical steps? Here are five related questions that build on this information:

1. What are the exact steps to calculate the slope from the angle \( 40^\circ \)?
2. How do you substitute the known points into the quadratic equation?
3. How can the tangent of the angle help derive the quadratic equation?
4. What are the next steps after determining the coefficients \( a \), \( b \), and \( c \)?
5. How does the shape of a quadratic function relate to the physical model of the bobsled track?

**Tip:** When solving real-world problems, always relate your mathematical model to the physical situation to ensure accuracy.

Frau Schlitter was on summer vacation at Area 47, a leisure park, and experiences the adrenaline rush of a 20-meter-high 'bob sled track.' The goal is to model the bob sled track using a quadratic function. The starting point height is (0, 20), and the lowest point is (30, 0). You are also given a 40° angle at the top of the ramp to help determine the slope.

In this problem, you'll model a bob sled track using a quadratic function. Learn how to use given points, slopes derived from angles, and algebraic techniques to describe the curve of the track. Includes calculations for finding the slope from a 40° angle and solving for a, b, and c in the quadratic equation. Suitable for high school students (Grades 9-12).

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