It costs 4 dollars to manufacture and distribute a backpack. If the backpacks sell at $x$ dollars each, the number sold, $n$, is given by $n=\frac{9}{x-4}+4(100-x)$. Find the selling price that will maximize profit.

The selling price that will maximize profit is $\$ \square$.
(Simplify your answer.)

Step 1 ：Let's denote the selling price as \(x\) and the number of backpacks sold as \(n\). The number of backpacks sold is given by the equation \(n = -4x + 400 + \frac{9}{x - 4}\).

Step 2 ：The profit \(P\) is given by the equation \(P = x(-4x + 400 + \frac{9}{x - 4}) - 4(-4x + 400 + \frac{9}{x - 4})\).

Step 3 ：To find the selling price that will maximize profit, we need to find the derivative of the profit function with respect to \(x\), set it equal to zero, and solve for \(x\). This will give us the critical points of the function.

Step 4 ：The derivative of the profit function is \(P' = x(-4 - \frac{9}{(x - 4)^2}) - 4x + 416 + \frac{9}{x - 4} + \frac{36}{(x - 4)^2}\).

Step 5 ：Solving \(P' = 0\) gives us the critical points. In this case, the critical point is \(x = 52\).

Step 6 ：We then need to determine which of these critical points is a maximum. We can do this by taking the second derivative of the profit function and evaluating it at the critical points.

Step 7 ：The second derivative of the profit function is \(P'' = \frac{18x}{(x - 4)^3} - 8 - \frac{18}{(x - 4)^2} - \frac{72}{(x - 4)^3}\).

Step 8 ：Evaluating \(P''\) at \(x = 52\) gives a negative value, which means that \(x = 52\) is a maximum.

Step 9 ：Thus, the selling price that will maximize profit is \(\boxed{\$52}\).