GED Mathematical Reasoning TestAlgebraic Problem Solving with Graphs and FunctionsHard
A production line's efficiency (E), in units per hour, is modeled by the function E(t) = -0.5t^2 + 6t + 10, where 't' is the number of hours since the start of the shift (0 ≤ t ≤ 8). At what time 't' is the efficiency at its maximum?
- A6 hours
- B10 hours
- C0 hours
- D8 hours
Show answer & explanationAnswer & explanation
Correct answer: A. 6 hours
The function E(t) = -0.5t^2 + 6t + 10 is a quadratic function opening downwards (because a=-0.5 < 0), so its maximum occurs at the vertex. The t-coordinate of the vertex is given by t = -b / (2a). Here, a = -0.5 and b = 6. So, t = -6 / (2 * -0.5) = -6 / -1 = 6 hours. This value is within the practical domain [0, 8].
Why the other options are wrong
- B. This value is outside the given domain of 0 ≤ t ≤ 8 hours.
- C. This is the efficiency at the start of the shift, not necessarily the maximum.
- D. This is the end of the shift, and while it could be the maximum if the vertex was outside the domain, the vertex is within it.
Vertex of a Parabola (Maximum/Minimum)
For a quadratic function in standard form f(x) = ax^2 + bx + c, the vertex is the point where the parabola reaches its maximum (if a < 0) or minimum (if a > 0) value. The x-coordinate of the vertex is given by x = -b / (2a).
- If 'a' is negative, the parabola opens downwards, and the vertex is a maximum.
- If 'a' is positive, the parabola opens upwards, and the vertex is a minimum.
- The y-coordinate of the vertex is found by substituting the x-coordinate back into the function.
Memory trick: Vertex 'X' is '-B over 2A'!