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The problem provides the function that models the approximate height of the Gateway Arch as:

\[
f(x) = -0.00635x^2 + 4x
\]

where \( x \) represents the distance from one end of the base to the other in feet, and \( f(x) \) gives the height in feet.

### To find the height at the tallest point:
The tallest point occurs at the vertex of the parabola, which is modeled by a quadratic function. For a quadratic function \( f(x) = ax^2 + bx + c \), the x-coordinate of the vertex is given by:

\[
x = -\frac{b}{2a}
\]

In this case:
- \( a = -0.00635 \)
- \( b = 4 \)

Substituting these values into the vertex formula:

\[
x = -\frac{4}{2(-0.00635)} = \frac{4}{0.0127} \approx 314.96
\]

Now, substitute \( x = 314.96 \) into the function to find the height at this point:

\[
f(314.96) = -0.00635(314.96)^2 + 4(314.96)
\]

Let's calculate this.The height of the tallest point of the Gateway Arch is approximately 630 feet when rounded to the nearest whole number.

Would you like more details on any part of the solution, or have any other questions?

Here are 5 related questions to explore further:
1. How can you find the total width of the base of the arch using this quadratic equation?
2. What would happen to the height if the coefficient of \( x^2 \) was less negative?
3. How does the value of \( b \) in the function affect the shape of the parabola?
4. Could you find the horizontal distance from the vertex to the ends of the base of the arch?
5. What is the physical significance of the vertex in this quadratic model?

**Tip:** The vertex of a quadratic function is always the maximum or minimum point depending on the sign of the quadratic term \( a \).

Suppose the approximate height of the Gateway Arch can be modeled by the function f(x) = -0.00635x^2 + 4x, where x represents the distance from one end of the base of the arch to the other in feet. How high above the ground is the tallest point of the arch?

Learn how to find the height of the tallest point of the Gateway Arch using a quadratic function model. This problem explores the application of the vertex formula to quadratic equations and is suitable for students in Grades 9-12.

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