A civil engineer is designing a parabolic arch for a pedestrian bridge. The height (y) of the arch in meters, as a function of its horizontal distance (x) from one end in meters, is given by y = -0.1x² + 4x. What is the maximum height of the arch?
- A30 meters
- B20 meters
- C40 meters
- D10 meters
Show answer & explanationAnswer & explanation
Correct answer: C. 40 meters
The equation y = -0.1x² + 4x is a quadratic function representing a downward-opening parabola, so its vertex will give the maximum height. The x-coordinate of the vertex is given by -b/(2a). Here, a = -0.1 and b = 4. So, x = -4 / (2 * -0.1) = -4 / -0.2 = 20. This is the horizontal distance at which the maximum height occurs. To find the maximum height, substitute x=20 back into the equation: y = -0.1(20)² + 4(20) = -0.1(400) + 80 = -40 + 80 = 40 meters.
Why the other options are wrong
- A. This could be a result of calculation errors during substitution or in the vertex formula.
- B. This is the x-coordinate of the vertex, not the maximum height (y-value).
- D. This is too low; it might be an error in calculating the vertex or a misinterpretation.
Quadratic Vertex (Max Height)
For a parabolic path or shape modeled by a quadratic function y = ax² + bx + c, the maximum height (or depth) is the y-coordinate of the vertex. The x-coordinate of the vertex, -b/(2a), gives the horizontal position where this maximum occurs.
- Parabola opens down if 'a' is negative (max height).
- Vertex x-coordinate = -b/(2a).
- Vertex y-coordinate (max height) = f(-b/(2a)).
Memory trick: The vertex is the arch's highest point, both left-right and up-down.