GED Mathematical Reasoning TestAlgebraic Problem Solving with Expressions and EquationsMedium
A chemist is observing a reaction where the concentration of a reactant, C, in moles per liter, changes over time, t, in minutes. The change is modeled by the function C(t) = 0.5t² - 3t + 8. At what time, in minutes, is the concentration at its minimum value?
- A0 minutes
- B3 minutes
- C1 minute
- D6 minutes
Show answer & explanationAnswer & explanation
Correct answer: B. 3 minutes
The function C(t) = 0.5t² - 3t + 8 is a quadratic function, representing a parabola. Since the coefficient of t² (a = 0.5) is positive, the parabola opens upwards, meaning its vertex represents the minimum value. The t-coordinate of the vertex is given by the formula t = -b/(2a). Here, a = 0.5 and b = -3. So, t = -(-3) / (2 * 0.5) = 3 / 1 = 3 minutes. This is the time at which the minimum concentration occurs.
Why the other options are wrong
- A. This is the initial time, but not necessarily the time of minimum concentration.
- C. This might be a result of calculation errors in the vertex formula.
- D. This could be a result of calculation errors, perhaps in the sign of 'b' or 'a'.
Quadratic Function Minimum
For a quadratic function f(x) = ax² + bx + c, if the leading coefficient 'a' is positive, the parabola opens upwards, and the vertex represents the minimum value of the function.
- Minimum occurs at the vertex x-coordinate: x = -b/(2a).
- Substitute this x-value into the function to find the minimum y-value.
- A positive 'a' indicates a minimum, a negative 'a' indicates a maximum.
Memory trick: Vertex is the bottom of the bowl, find its time coordinate.