1. **State the problem:** We want to find the number of cookies $x$ Nancy must sell to maximize her monthly profit given by the function $$P(x) = 50\sqrt{x} - 0.5x - 75.$$\n\n2. **Formula and rules:** To find the maximum of a function, we use calculus by finding the critical points where the derivative $P'(x)$ is zero or undefined, and then check which point gives a maximum.\n\n3. **Find the derivative:**\n$$P(x) = 50x^{1/2} - 0.5x - 75$$\nUsing the power rule,\n$$P'(x) = 50 \cdot \frac{1}{2} x^{-1/2} - 0.5 = \frac{25}{\sqrt{x}} - 0.5.$$\n\n4. **Set the derivative equal to zero to find critical points:**\n$$\frac{25}{\sqrt{x}} - 0.5 = 0$$\n$$\frac{25}{\sqrt{x}} = 0.5$$\nMultiply both sides by $\sqrt{x}$:\n$$25 = 0.5 \sqrt{x}$$\nDivide both sides by 0.5:\n$$\frac{25}{0.5} = \sqrt{x}$$\n$$50 = \sqrt{x}$$\n\n5. **Solve for $x$:**\nSquare both sides:\n$$x = 50^2 = 2500.$$\n\n6. **Check if this critical point is a maximum:**\nCalculate the second derivative:\n$$P''(x) = -\frac{25}{2} x^{-3/2}.$$\nAt $x=2500$,\n$$P''(2500) = -\frac{25}{2} (2500)^{-3/2} < 0,$$\nwhich means the function is concave down and $x=2500$ is a maximum point.\n\n**Final answer:** Nancy must sell **2500 cookies** per month to maximize her profit.