To find the maximum profit for the bike company, we first need to determine the profit function $P(x)$, which is the difference between the revenue $R(x)$ and the cost $C(x)$. Given:

$R(x) = 325x - 2x^2$ $C(x) = 4500 + 55x$

The profit function $P(x)$ is: $P(x) = R(x) - C(x) = (325x - 2x^2) - (4500 + 55x)$

Simplifying this, we get: $P(x) = -2x^2 + 325x - 55x - 4500$ $P(x) = -2x^2 + 270x - 4500$

To find the maximum profit, we need to find the vertex of the parabola represented by $P(x)$. The $x$-coordinate of the vertex (and thus the number of bikes sold for maximum profit) can be found using the formula $x = -\frac{b}{2a}$, where $a$ and $b$ are the coefficients from the quadratic term and the linear term of $P(x)$ respectively.

Here, $a = -2$ and $b = 270$, so: $x = -\frac{270}{2(-2)} = \frac{270}{4} = 67.5$

Since the number of bikes sold, $x$, must be an integer, we will evaluate the profit at $x = 67$ and $x = 68$ to determine which gives a higher profit.

Calculating $P(67)$: $P(67) = -2(67)^2 + 270(67) - 4500$

Calculating $P(68)$: $P(68) = -2(68)^2 + 270(68) - 4500$

We will now compute these values.