Answer :
To determine the equation of the cross-sectional parabola of a dome that is 200 feet in diameter with a maximum height of 50 feet, follow these steps:
1. Understanding the Symmetry of the Parabola:
- The parabola is symmetric about the vertical axis passing through the vertex.
- The vertex of the parabola is at the maximum height of the dome. Therefore, the vertex is (0, 50).
2. Equation of Parabola:
- Since the parabola opens downwards, its standard form is \( y = ax^2 + b \).
- Its vertex form is \( y = a(x - h)^2 + k \), where (h, k) is the vertex of the parabola.
3. Substitute the Vertex:
- From the above problem, the vertex is at (0, 50).
- Therefore, the equation becomes \( y = a(x - 0)^2 + 50 \) or simplified to \( y = ax^2 + 50 \).
4. Determine \( a \):
- The parabola passes through the point where the diameter reaches its width, which is at \( x = \pm 100 \) since the radius is half the diameter.
- When \( x = 100 \), \( y = 0 \) (the height of the dome at the edges is zero).
- Substitute \( x = 100 \) and \( y = 0 \) into the equation \( 0 = a(100)^2 + 50 \):
[tex]\[ 0 = 10000a + 50 \][/tex]
- Solve for \( a \):
[tex]\[ -50 = 10000a \Rightarrow a = -\frac{50}{10000} = -\frac{1}{200} \][/tex]
5. Form the Parabola Equation:
- Substituting \( a = -\frac{1}{200} \) back into the equation:
[tex]\[ y = -\frac{1}{200}x^2 + 50 \][/tex]
6. Compare with Options:
- The provided options are:
[tex]\[ \text{A. } y = -\frac{x^2}{400} + \frac{5,000}{400} \][/tex]
[tex]\[ \text{B. } y = -\frac{x^2}{200} + \frac{10,000}{200} \][/tex]
[tex]\[ \text{C. } y = -\frac{x^2}{400} + \frac{5,000}{50} \][/tex]
[tex]\[ \text{D. } y = -\frac{x^2}{400} + \frac{5,000}{2,000} \][/tex]
- The correct form derived is:
[tex]\[ y = -\frac{x^2}{200} + 50 = -\frac{x^2}{200} + \frac{10,000}{200} \][/tex]
Therefore, the correct answer is:
\[
\boxed{B}
1. Understanding the Symmetry of the Parabola:
- The parabola is symmetric about the vertical axis passing through the vertex.
- The vertex of the parabola is at the maximum height of the dome. Therefore, the vertex is (0, 50).
2. Equation of Parabola:
- Since the parabola opens downwards, its standard form is \( y = ax^2 + b \).
- Its vertex form is \( y = a(x - h)^2 + k \), where (h, k) is the vertex of the parabola.
3. Substitute the Vertex:
- From the above problem, the vertex is at (0, 50).
- Therefore, the equation becomes \( y = a(x - 0)^2 + 50 \) or simplified to \( y = ax^2 + 50 \).
4. Determine \( a \):
- The parabola passes through the point where the diameter reaches its width, which is at \( x = \pm 100 \) since the radius is half the diameter.
- When \( x = 100 \), \( y = 0 \) (the height of the dome at the edges is zero).
- Substitute \( x = 100 \) and \( y = 0 \) into the equation \( 0 = a(100)^2 + 50 \):
[tex]\[ 0 = 10000a + 50 \][/tex]
- Solve for \( a \):
[tex]\[ -50 = 10000a \Rightarrow a = -\frac{50}{10000} = -\frac{1}{200} \][/tex]
5. Form the Parabola Equation:
- Substituting \( a = -\frac{1}{200} \) back into the equation:
[tex]\[ y = -\frac{1}{200}x^2 + 50 \][/tex]
6. Compare with Options:
- The provided options are:
[tex]\[ \text{A. } y = -\frac{x^2}{400} + \frac{5,000}{400} \][/tex]
[tex]\[ \text{B. } y = -\frac{x^2}{200} + \frac{10,000}{200} \][/tex]
[tex]\[ \text{C. } y = -\frac{x^2}{400} + \frac{5,000}{50} \][/tex]
[tex]\[ \text{D. } y = -\frac{x^2}{400} + \frac{5,000}{2,000} \][/tex]
- The correct form derived is:
[tex]\[ y = -\frac{x^2}{200} + 50 = -\frac{x^2}{200} + \frac{10,000}{200} \][/tex]
Therefore, the correct answer is:
\[
\boxed{B}
