GED Mathematical Reasoning TestAlgebraic Problem Solving with Expressions and EquationsMedium
A scientist is studying the relationship between the dosage of a new fertilizer (d, in grams) and the growth of a plant (G, in centimeters). The growth is modeled by the function G(d) = -0.1d^2 + 2d + 5. What is the optimal dosage (in grams) to achieve maximum growth?
- A15 grams
- B10 grams
- C5 grams
- D20 grams
Show answer & explanationAnswer & explanation
Correct answer: B. 10 grams
The growth function is a quadratic equation, representing a parabola opening downwards. The optimal dosage for maximum growth occurs at the x-coordinate (in this case, 'd') of the vertex of the parabola.
Why the other options are wrong
- A. This is a plausible distractor, possibly from an arithmetic error.
- C. This is a plausible distractor, possibly from an arithmetic error.
- D. This value would result in zero or negative growth, indicating it's far past the optimal point.
Quadratic Vertex (Optimal Input)
For a quadratic function f(x) = ax^2 + bx + c, the x-coordinate of the vertex, x = -b / (2a), represents the optimal input value (e.g., dosage, price) that leads to the maximum or minimum output (e.g., growth, profit).
- Used when a problem asks for the input that maximizes or minimizes an output.
- The sign of 'a' determines if it's a maximum (a<0) or minimum (a>0).
- The vertex formula is directly used to find this optimal input.
Memory trick: Vertex Finds the 'X' Factor.