A record club has found that the marginal profit, $P^{\prime}(x)$, in cents, is given by $P^{\prime}(x)=-0.0008 x^{3}+0.25 x^{2}+57.6 x$ for $x \leq 300$, where $x$ is the number of members currently enrolled in the club. Approximate the total profit when 180 members are enrolled by computing the sum $\sum_{i=1}^{6} P^{\prime}\left(x_{i}\right) \Delta x$ with $\Delta \mathrm{x}=30$.
The total profit when 180 members are enrolled is approximately $\$ 10030.50$. (Round to the nearest cent as needed.)

Step 1 ：The problem is asking for the total profit when 180 members are enrolled. This can be approximated by computing the sum \(\sum_{i=1}^{6} P^{\prime}\left(x_{i}\right) \Delta x\) with \(\Delta x=30\). This is a Riemann sum, which is a method for approximating the total value of a function over an interval. In this case, the function is the marginal profit function \(P^{\prime}(x)\), and the interval is from 1 to 180 (the number of members). The \(\Delta x\) value of 30 represents the width of each subinterval in the Riemann sum. The \(x_i\) values are the points in each subinterval at which the function is evaluated.

Step 2 ：To compute the Riemann sum, we need to evaluate the function \(P^{\prime}(x)\) at each \(x_i\) and multiply by \(\Delta x\), then sum these values. The \(x_i\) values are [30, 60, 90, 120, 150, 180].

Step 3 ：The total profit calculated is in cents, so we need to convert it to dollars by dividing by 100. The total profit when 180 members are enrolled is approximately \$14171.22.

Step 4 ：\(\boxed{\text{The total profit when 180 members are enrolled is approximately \$14171.22.}}\)