To determine the selling price that maximizes profit, we need to analyze the given profit function:
[tex]\[ P(x) = -25x^2 + 1600x - 3600 \][/tex]
This function is a quadratic equation in the form [tex]\( ax^2 + bx + c \)[/tex], with:
[tex]\[ a = -25, \, b = 1600, \, \text{and} \, c = -3600 \][/tex]
For quadratic functions of the form [tex]\( ax^2 + bx + c \)[/tex], the vertex (which represents the maximum or minimum point) occurs at:
[tex]\[ x = -\frac{b}{2a} \][/tex]
Here, the coefficient [tex]\( a \)[/tex] is negative ([tex]\( -25 \)[/tex]), indicating that the parabola opens downwards, thus the vertex represents the maximum profit point.
Plug the values from the profit function into the vertex formula:
[tex]\[ x = -\frac{1600}{2 \cdot (-25)} \][/tex]
Simplify the expression:
[tex]\[ x = -\frac{1600}{-50} \][/tex]
[tex]\[ x = 32 \][/tex]
Thus, the selling price that gives the maximum profit is:
[tex]\[ \boxed{32} \][/tex]
This means the company will earn the maximum profit when each spy camera is sold at [tex]\( \$32 \)[/tex].
