Digital SATMath: AlgebraHard
A civil engineer is designing a suspension bridge. The height, h(x), in meters, of the main cable above the bridge deck can be modeled by the function h(x) = 0.005x^2 - 0.6x + 20, where x is the horizontal distance in meters from the left support. What is the minimum height of the cable above the deck?
- A11 meters
- B5 meters
- C2 meters
- D20 meters
Show answer & explanationAnswer & explanation
Correct answer: B. 5 meters
The function h(x) = 0.005x^2 - 0.6x + 20 is a quadratic function in the form ax^2 + bx + c. Since a = 0.005 > 0, the parabola opens upwards, and its vertex represents the minimum height. The x-coordinate of the vertex is given by x = -b / (2a). Here, x = -(-0.6) / (2 * 0.005) = 0.6 / 0.01 = 60. Substitute x = 60 back into the function: h(60) = 0.005(60)^2 - 0.6(60) + 20 = 0.005(3600) - 36 + 20 = 18 - 36 + 20 = 2. So the minimum height is 2 meters.
Why the other options are wrong
- A. This might be the result of calculation error or confusion with the coefficients.
- C. This is incorrect, likely a calculation error after finding the vertex's x-coordinate.
- D. This is the y-intercept (c value), not the minimum height, which occurs at the vertex.
Minimum of Quadratic Function
For a quadratic function in the form f(x) = ax^2 + bx + c, if a > 0, the parabola opens upwards, and its vertex represents the minimum value of the function.
- The x-coordinate of the vertex is given by x = -b / (2a).
- The minimum value is the y-coordinate of the vertex, f(-b / (2a)).
- Relevant for optimization problems like finding lowest point or minimum cost.
Memory trick: Vertex is the valley (or peak) of the parabola.