https://file.aiquickdraw.com/mathbot/file/1726379859972a0on28tp.jpg

Let's analyze the problem provided in the image:

We are dealing with a quadratic curve modeling the Lowry Avenue Bridge. The important information is as follows:

- The quadratic function is of the form \( f(x) = ax^2 + bx + c \).
- The endpoints of the bridge are at \( (0, 0) \) and \( (480, 0) \), and the vertex is at \( (240, 33) \).

### Part (a)(i): 
We are first asked to find the value of \( c \). Since \( f(0) = 0 \) (the curve passes through the origin), this implies:
\[
c = 0
\]

### Part (a)(ii): 
Next, we need to find the equation of the quadratic curve.

We can use the vertex form of a quadratic equation:
\[
f(x) = a(x - h)^2 + k
\]
where \( (h, k) \) is the vertex. Here, the vertex is \( (240, 33) \), so:
\[
f(x) = a(x - 240)^2 + 33
\]
We also know that the curve passes through \( (0, 0) \), so we can substitute \( (0, 0) \) into the equation to find \( a \):
\[
0 = a(0 - 240)^2 + 33
\]
\[
0 = a(57600) + 33
\]
\[
a = -\frac{33}{57600} = -\frac{11}{19200}
\]
Thus, the equation of the quadratic curve is:
\[
f(x) = -\frac{11}{19200}(x - 240)^2 + 33
\]

### Part (b): 
The bridge is supported by 7 steel beams that are evenly spaced. To find the height of the shortest beam, we need to calculate the height of the beam at the position closest to either end. Since the beams are evenly spaced along the \( x \)-axis, the first beam from the left is at \( x = 480/7 = 68.57 \).

We can plug this value into the quadratic equation to find the height of the beam:
\[
f(68.57) = -\frac{11}{19200}(68.57 - 240)^2 + 33
\]

Let me calculate this height.The height of the shortest steel beam, located at \( x = 68.57 \), is approximately 16.16 meters.

Let me know if you'd like any further clarification or details.

### Relative Questions:
1. Why do we use the vertex form of the quadratic equation in this problem?
2. How would the quadratic equation change if the vertex had different coordinates?
3. What happens to the shape of the bridge if the coefficient \( a \) becomes positive?
4. Can we use symmetry to find the height of other beams on the bridge?
5. How would the model change if the endpoints of the bridge were not at \( (0, 0) \) and \( (480, 0) \)?

### Tip:
When solving quadratic problems, using the vertex form of the equation simplifies the process, especially when the vertex is given directly in the problem.

The Lowry Avenue Bridge is modeled as a quadratic curve, and you are asked to find the equation of the curve, along with the height of the shortest steel beam supporting the bridge. The key information provided includes the coordinates of the endpoints of the bridge (0,0) and (480,0), with a vertex at (240,33). The equation is in the form f(x) = ax^2 + bx + c.

This problem involves finding the quadratic equation that models the Lowry Avenue Bridge. The curve has endpoints at (0,0) and (480,0) with a vertex at (240,33). We also calculate the height of the shortest steel beam supporting the bridge, using algebraic methods.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation