https://r2.erweima.ai/mathbot/image/1726965817792xm1g0v71.jpg

The uploaded image explains the derivation of the equation for a parabolic curve and the calculation of a cable's height when \( x = 1000 \). Let's walk through the steps in detail:

### Explanation

1. **Standard Form of Parabola**: 
   The equation of a parabola is given as:
   \[
   x^2 = 4py
   \]

2. **Using Points on the Parabola**:
   The points \((-2100, 526)\) and \((2100, 526)\) lie on the parabola. Substituting the point \((2100, 526)\) into the equation:
   \[
   (2100)^2 = 4p(526)
   \]
   Solving for \(4p\):
   \[
   4p = \frac{(2100)^2}{526}
   \]

3. **Substituting \(4p\) back into the general equation**:
   The parabola's equation now becomes:
   \[
   x^2 = \frac{(2100)^2}{526} \cdot y
   \]

4. **Finding the height of the cable at \(x = 1000\)**:
   Substitute \(x = 1000\) into the equation:
   \[
   (1000)^2 = \frac{(2100)^2}{526} \cdot y
   \]
   Solving for \(y\):
   \[
   y = \frac{526 \cdot (1000)^2}{(2100)^2} \approx 119.3
   \]

Hence, the height of the cable when \( x = 1000 \) is approximately \( y \approx 119.3 \) units.

Would you like more details or have any questions?

### Five Related Questions
1. How do we derive the general equation of a parabola?
2. What is the significance of the parameter \(p\) in a parabolic equation?
3. How would the height change if a different \( x \)-value were used?
4. Can you explain the relationship between the focal length and the points used here?
5. How do parabolas differ from other conic sections like ellipses and hyperbolas?

### Tip
When dealing with parabolas, always check if the points you have lie symmetrically about the y-axis, which indicates a standard form with vertex at the origin.

Given the equation of a parabola and points on the parabola, how can we find the height of the cable at a specific x-coordinate?

Learn how to derive the equation of a parabola and calculate the height of a cable at a specific x-coordinate using points on the curve. This problem involves key concepts in conic sections and algebra, including solving for the height y when x = 1000.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation