The problem involves finding a quadratic function to model the cross section of a tunnel entrance, which has a parabolic shape. Here’s the breakdown:

The information given:

1. The maximum height of the tunnel is $9.8$ meters.
2. The maximum width of the tunnel is also $9.8$ meters.

Since the tunnel has a parabolic shape, the quadratic function representing its cross section can be written in vertex form:

$y = a(x - h)^2 + k$

where:

• $h$ and $k$ are the constants representing the coordinates of the vertex (the maximum point in this case).
• $a$ is a scaling factor that defines the "width" or "spread" of the parabola.

Step-by-Step Solution:

1. Identify the Vertex:

   • The maximum height (vertex) of the tunnel is 9.8 meters, so $k = 9.8$.
   • Since the tunnel's width is also 9.8 meters, this width would span symmetrically around the central vertical axis. Therefore, the vertex is at $x = 0$, making $h = 0$.

   This gives the function form: $y = a(x - 0)^2 + 9.8 \Rightarrow y = ax^2 + 9.8$

2. Determine the Value of $a$:

   • The tunnel width implies that at a distance of $x = 4.9$ meters on either side of the center, the height should be zero (since 9.8 meters is the total width, each half of the width is 4.9 meters).

   • Substitute $x = 4.9$ and $y = 0$ (height is zero at the edges) into the equation to solve for $a$:

     $0 = a(4.9)^2 + 9.8$

     $-9.8 = a \times 24.01$

     $a = \frac{-9.8}{24.01} \approx -0.408$

3. Final Equation:

   • Substitute $a$ back into the equation:

     $y = -0.408x^2 + 9.8$

This is the quadratic function that models the cross section of the tunnel entrance.

Would you like a detailed explanation of any part, or do you have further questions?

Related Questions:

1. How does the value of $a$ affect the width of the parabola?
2. What would happen to the equation if the maximum height changed?
3. How could this model be adapted for tunnels with different shapes?
4. What are other real-world applications of parabolic models?
5. How would the model change if the tunnel cross section was elliptical?

Tip:

Always check boundary points (like the maximum width edges here) to ensure the model accurately represents real-world constraints.