https://file.aiquickdraw.com/mathbot/file/1725617992171rex1cprb.jpg

Answer the math problem and provide the ff. Concavity: Equation: Vertex: Value of c: Focus: Directrix: Length of Latus Rectum: Endpoints of LR: Principal Axis:

Let's start by analyzing the problem. The given image describes a suspension bridge's cable, which forms the shape of a parabola. The towers supporting the cable are 800 feet apart and 250 feet high. The cable's lowest point is 20 feet above the ground, and we are asked to determine several properties of the parabola.

### Step-by-Step Solution:

1. **Concavity**:
   The parabola opens upwards since it represents the shape of a suspension cable hanging between two towers.

2. **Equation of the Parabola**:
   Let's establish a coordinate system where the vertex of the parabola is at the lowest point of the cable, 20 feet above the ground. Thus, the vertex will be at \((0, 20)\), assuming the origin is at the middle of the bridge. The two towers are located at \(x = -400\) feet and \(x = 400\) feet, each at a height of 250 feet.

   Since the parabola opens upward, we can write the equation in the standard form:
   \[
   y = ax^2 + bx + c
   \]
   Given that the vertex \((h, k) = (0, 20)\), the equation simplifies to:
   \[
   y = a(x - 0)^2 + 20 \implies y = ax^2 + 20
   \]

   To find \(a\), we use the point \((400, 250)\):
   \[
   250 = a(400)^2 + 20
   \]
   \[
   230 = 160000a \implies a = \frac{230}{160000} = \frac{23}{16000} = 0.0014375
   \]
   
   Thus, the equation of the parabola is:
   \[
   y = 0.0014375x^2 + 20
   \]

3. **Vertex**:
   The vertex of the parabola is at \((0, 20)\).

4. **Value of c**:
   In the standard quadratic form \(y = ax^2 + bx + c\), \(c = 20\), representing the y-intercept (the height of the lowest point of the cable).

5. **Focus**:
   The focus of a parabola in the form \(y = ax^2 + c\) is given by the formula:
   \[
   (h, k + \frac{1}{4a})
   \]
   Here, \(h = 0\), \(k = 20\), and \(a = 0.0014375\), so:
   \[
   \text{Focus} = \left(0, 20 + \frac{1}{4 \times 0.0014375}\right) = \left(0, 20 + 173.913\right) \approx (0, 193.913)
   \]

6. **Directrix**:
   The directrix of a parabola is given by:
   \[
   y = k - \frac{1}{4a}
   \]
   Substituting the values:
   \[
   \text{Directrix} = 20 - 173.913 = -153.913
   \]

7. **Length of the Latus Rectum**:
   The length of the latus rectum is \(\frac{1}{|a|}\):
   \[
   \text{Length of Latus Rectum} = \frac{1}{0.0014375} \approx 695.65
   \]

8. **Endpoints of Latus Rectum**:
   The endpoints of the latus rectum are given by:
   \[
   \left(h \pm \frac{1}{2a}, k + \frac{1}{4a}\right)
   \]
   Substituting the values:
   \[
   \left(0 \pm \frac{1}{2 \times 0.0014375}, 193.913\right) = (348.82, 193.913), (-348.82, 193.913)
   \]

9. **Principal Axis**:
   The principal axis of a parabola is the line passing through the vertex and focus, which in this case is the y-axis or \(x = 0\).

### Summary of Answers:
- **Concavity**: Upward
- **Equation**: \(y = 0.0014375x^2 + 20\)
- **Vertex**: \((0, 20)\)
- **Value of c**: 20
- **Focus**: \((0, 193.913)\)
- **Directrix**: \(y = -153.913\)
- **Length of Latus Rectum**: 695.65
- **Endpoints of LR**: \((348.82, 193.913), (-348.82, 193.913)\)
- **Principal Axis**: \(x = 0\)

Would you like more details or have any questions?

### Related Questions to Explore:
1. How does the value of \(a\) affect the shape of the parabola?
2. How can you derive the equation of a parabola given different sets of points?
3. What are the applications of parabolic equations in real life?
4. How would the equation change if the vertex were at a different point?
5. How do you find the intersection points of a parabola with a line?

### Tip:
When working with parabolic equations, always check for symmetry and vertex location to simplify the calculations.

The cable of a suspension bridge forms the shape of a parabola. The towers supporting the cable are 800 feet apart and 250 feet high. What is the height of the cable at a point 60 feet from the center of the bridge? Assume that the lowest point of the cable is 20 feet from the ground.

Solve the parabola equation for a suspension bridge, where the towers are 800 feet apart and 250 feet high. Learn how to find the height of the cable 60 feet from the center, and explore the focus, directrix, vertex, and other key parabola properties.

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