Digital SATMath: AlgebraHard
A civil engineer is designing a parabolic arch for a bridge. The height of the arch, h(x), in meters, at a horizontal distance x meters from the left support, is given by h(x) = -0.05x(x - 60). What is the maximum height of the arch?
- A45 meters
- B75 meters
- C60 meters
- D30 meters
Show answer & explanationAnswer & explanation
Correct answer: A. 45 meters
The equation h(x) = -0.05x(x - 60) is a quadratic function in factored form. The roots (x-intercepts) are x = 0 and x = 60. The x-coordinate of the vertex (where the maximum height occurs) is exactly halfway between the roots: x = (0 + 60) / 2 = 30. Now, substitute x = 30 into the function to find the maximum height: h(30) = -0.05(30)(30 - 60) = -0.05(30)(-30) = -0.05(-900) = 45 meters.
Why the other options are wrong
- B. This value might result from calculation errors or incorrect interpretation of the vertex.
- C. This is one of the x-intercepts, not the maximum height.
- D. This is the x-coordinate of the vertex, not the maximum height.
Maximum of Quadratic (Factored Form)
For a quadratic function in factored form y = a(x - r1)(x - r2), the x-coordinate of the vertex occurs halfway between the roots r1 and r2. The y-coordinate at this point is the maximum or minimum.
- Roots are the x-intercepts where y=0.
- X-coordinate of vertex: (r1 + r2) / 2.
- Substitute x-vertex into function to find max/min y-value.
Memory trick: Factored Max: Roots are easy, middle is 'x', plug it in to get 'y' high!