GED Mathematical Reasoning TestAlgebraic Problem Solving with Expressions and EquationsMedium
A civil engineer is calculating the sag in a suspension cable. The sag (S) in meters at a distance 'x' meters from the left support is approximated by the function S(x) = 0.005x^2 - 0.2x + 5. What is the minimum sag of the cable?
- A2 meters
- B4 meters
- C5 meters
- D3 meters
Show answer & explanationAnswer & explanation
Correct answer: D. 3 meters
The function S(x) = 0.005x^2 - 0.2x + 5 is a quadratic with a positive leading coefficient (0.005), so it has a minimum. The x-coordinate of the vertex is x = -b/(2a) = -(-0.2)/(2*0.005) = 0.2/0.01 = 20. Substitute x=20 into the function: S(20) = 0.005(20)^2 - 0.2(20) + 5 = 0.005(400) - 4 + 5 = 2 - 4 + 5 = 3.
Why the other options are wrong
- A. This is incorrect; it might be a calculation error (e.g., 2-4+5 = 3, not 2).
- B. This is an incorrect calculation or misinterpretation of the vertex formula.
- C. This is the y-intercept, which represents the sag at the left support (x=0), not the minimum sag.
Quadratic Function Minimum
For a quadratic function f(x) = ax^2 + bx + c, if 'a' is positive, the parabola opens upwards, and its vertex represents the minimum value of the function.
- Occurs at the vertex, where x = -b/(2a).
- The minimum value is f(-b/(2a)).
- Relevant in problems seeking lowest cost, shortest distance, or minimum value.
Memory trick: Minimum: 'a' is positive, find the valley's lowest point!