Westonci.ca is the ultimate Q&A platform, offering detailed and reliable answers from a knowledgeable community. Experience the ease of finding accurate answers to your questions from a knowledgeable community of professionals. Discover detailed answers to your questions from a wide network of experts on our comprehensive Q&A platform.
Sagot :
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}
Thank you for your visit. We are dedicated to helping you find the information you need, whenever you need it. Thanks for stopping by. We strive to provide the best answers for all your questions. See you again soon. We're here to help at Westonci.ca. Keep visiting for the best answers to your questions.