https://file.aiquickdraw.com/mathbot/file/1725381658190vcmbj0tg.jpg

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### Image Description:
The image shows a question involving the Sydney Harbour Bridge. The bridge's arch can be modeled by a quadratic equation in the form \( y = -0.0032x^2 + 0.52x \), where \( x \) is the horizontal distance in meters from the left pylon, and \( y \) is the height in meters of the arch above the base of the bridge. The bridge is labeled with a pylon on each side, and the span between the pylons is marked as 52 meters.

### Problem Summary:
We are asked to find the maximum height of the arch above the base of the bridge. 

Let's solve the problem step-by-step. The equation of the arch is a quadratic equation of the form \( ax^2 + bx + c \). The maximum height occurs at the vertex of the parabola. For a quadratic equation in the form \( y = ax^2 + bx + c \), the x-coordinate of the vertex can be found using the formula:

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

Substitute the values of \( a = -0.0032 \) and \( b = 0.52 \) to find the x-coordinate of the maximum point. Then, substitute this x-value back into the original equation to find the corresponding height \( y \). I'll do the calculations now.It seems like I can’t do more advanced data analysis right now. Please try again later. However, I can still help you solve the problem manually.

### Solution:
To find the maximum height of the arch, we use the given quadratic equation:

\[
y = -0.0032x^2 + 0.52x
\]

The x-coordinate of the vertex (where the maximum height occurs) can be calculated using the formula:

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

Here, \(a = -0.0032\) and \(b = 0.52\). Plugging in the values:

\[
x = -\frac{0.52}{2 \times (-0.0032)} = \frac{0.52}{0.0064} \approx 81.25
\]

Now, substitute \(x = 81.25\) back into the original equation to find the maximum height:

\[
y = -0.0032(81.25)^2 + 0.52 \times 81.25
\]

Calculating:

1. \(81.25^2 = 6601.56\)
2. \( -0.0032 \times 6601.56 = -21.125\)
3. \( 0.52 \times 81.25 = 42.25 \)

Thus,

\[
y = -21.125 + 42.25 = 21.125
\]

### Result:
The maximum height of the arch above the base of the bridge is **21.125 meters**.

Would you like to dive deeper into any part of this problem, or do you have any other questions? 

### Additional Questions:
1. How would you solve for the height at a different point along the bridge?
2. What is the significance of the coefficient of \(x^2\) in a quadratic equation?
3. How can we determine the width of the arch at a specific height?
4. What happens to the arch's equation if the bridge was constructed differently?
5. How would the equation change if we measured from the right pylon instead of the left?

### Tip:
For any quadratic equation, the vertex formula is crucial for finding the maximum or minimum point of the parabola.

The arch of the Sydney Harbour Bridge in Sydney, Australia, can be modeled by the equation y = -0.0032x^2 + 0.52x, where x is the distance in meters from the left pylon and y is the height in meters of the arch above the base. For what distance x is the arch above the base the tallest?

Learn how to determine the maximum height of the Sydney Harbour Bridge arch by solving a quadratic equation. This problem involves key algebraic concepts such as quadratic equations and the vertex formula. Ideal for students in Grades 9-11.

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