GED Mathematical Reasoning TestAlgebraic Problem Solving with Expressions and EquationsMedium
A biologist is studying the population of a certain species of fish in a lake. The population can be modeled by the function P(t) = -2t^2 + 80t + 1200, where P(t) is the population after 't' years. What is the maximum population the lake can sustain according to this model?
- A1600 fish
- B2400 fish
- C2000 fish
- D1200 fish
Show answer & explanationAnswer & explanation
Correct answer: C. 2000 fish
The population function is a quadratic equation, which represents a parabola opening downwards (due to the negative coefficient of t^2). The maximum population occurs at the vertex of this parabola.
Why the other options are wrong
- A. This is a plausible distractor, possibly from an arithmetic error in the vertex calculation.
- B. This value is too high and does not correspond to the vertex.
- D. This is the initial population at t=0, not the maximum.
Quadratic Function Maximum
For a quadratic function in the form f(x) = ax^2 + bx + c, if a < 0, the parabola opens downwards and has a maximum value at its vertex.
- The x-coordinate of the vertex is given by x = -b / (2a).
- The maximum value is the y-coordinate of the vertex, f(-b / (2a)).
- Used to find peak values in real-world scenarios.
Memory trick: Vertex Vigorously Varies Value.