
The parabolic arch shown in the figure is 50 feet above the water at the center and 200 feet wide at the base.Will a boat that is 30 feet tall clear the arch 30 feet from the center?

We are given a parabolic arch that is 50 feet above the water at its center and 200 feet wide at the base. We need to determine whether a boat that is 30 feet tall can clear the arch when it is 30 feet from the center.

### Step-by-step solution:

1. **Modeling the Parabola:**
   - Since the parabola is symmetric about the center, we place the vertex of the parabola at the origin, \( (0, 50) \), where 50 feet is the height of the arch at the center.
   - The arch is 200 feet wide, so the distance from the center to either side of the base is 100 feet (half of 200 feet).

   The equation of a vertical parabola is typically written as:
   \[
   y = a(x - h)^2 + k
   \]
   where \( (h, k) \) is the vertex. Here, \( h = 0 \) and \( k = 50 \), so the equation becomes:
   \[
   y = a x^2 + 50
   \]

2. **Using the Base of the Arch:**
   - At the base of the arch, \( y = 0 \) when \( x = \pm 100 \) feet (since the base is 200 feet wide).
   - Plugging \( y = 0 \) and \( x = 100 \) into the equation to find \( a \):
   \[
   0 = a(100)^2 + 50
   \]
   \[
   0 = 10000a + 50
   \]
   \[
   10000a = -50
   \]
   \[
   a = -\frac{50}{10000} = -\frac{1}{200}
   \]

   Thus, the equation of the parabola is:
   \[
   y = -\frac{1}{200}x^2 + 50
   \]

3. **Finding the Height of the Arch 30 Feet from the Center:**
   - We need to find the height of the arch at \( x = 30 \) feet.
   - Substituting \( x = 30 \) into the equation:
   \[
   y = -\frac{1}{200}(30)^2 + 50
   \]
   \[
   y = -\frac{1}{200}(900) + 50
   \]
   \[
   y = -4.5 + 50 = 45.5
   \]

4. **Comparing the Boat Height with the Arch Height:**
   - The height of the arch 30 feet from the center is 45.5 feet.
   - The boat is 30 feet tall.

Since 45.5 feet (the height of the arch) is greater than 30 feet (the height of the boat), the boat **will clear the arch**.

### Conclusion:
Yes, the boat will safely clear the arch with a margin of 15.5 feet (45.5 feet - 30 feet).

---

Do you want more details on any part of the solution or have any questions? 

Here are 5 questions related to this problem:
1. How would the problem change if the width of the arch were increased?
2. What if the boat was 40 feet tall instead of 30 feet tall? Would it still clear the arch?
3. Can you derive the equation of the parabola if the vertex is not at the origin?
4. How does changing the value of \( a \) affect the shape of the parabola?
5. How would you find the height of the arch at different points along the base?

**Tip:** In problems involving symmetry, using the vertex of the parabola as the origin can simplify calculations.

The parabolic arch shown in the figure is 50 feet above the water at the center and 200 feet wide at the base. Will a boat that is 30 feet tall clear the arch 30 feet from the center?

Learn how to determine whether a 30-foot tall boat can pass under a parabolic arch that is 50 feet high at the center and 200 feet wide. This solution uses the equation of a parabola and key geometric concepts, suitable for students in Grades 9-11.

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