The profit, [tex]P[/tex] dollars, earned by a company producing spy cameras is approximated to be [tex]P=-25x^2+1600x-3600[/tex], where [tex]x[/tex] dollars is the selling price of one spy camera.

What selling price gives the maximum profit?

A. [tex]$\$[/tex] 64[tex]$
B. $[/tex]\[tex]$ 32$[/tex]
C. [tex]$\$[/tex] 16[tex]$
D. $[/tex]\[tex]$ 25$[/tex]



Answer :

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].