Question

The following empirical relationship describes how the number of trees \( N(t) \) in a patch changes over time \( t \): \[ N(t) = -2t^2 + 12t + 24 \] where \( t = 0 \) is when the number of trees were first counted. Given this relationship, the maximum number of trees that occur in the patch is

Hint:

When dealing with quadratic functions, remember that the vertex represents either the maximum or minimum value depending on the direction of the parabola (opening up or down).

Answer 1

The equation provided is a quadratic equation in the form of \( N(t) = at^2 + bt + c \), where \( a = -2 \), \( b = 12 \), and \( c = 24 \). The maximum value for a downward opening parabola (since \( a<0 \)) occurs at the vertex. The \( t \)-coordinate of the vertex can be found using \( t = -\frac{b}{2a} \). Calculating the vertex: \[ t = -\frac{12}{2 \times -2} = 3 \] Substituting \( t = 3 \) back into the equation to find \( N(t) \): \[ N(3) = -2(3)^2 + 12 \times 3 + 24 = 42 \] Thus, the maximum number of trees in the patch is 42 (rounded to the nearest integer).