To approximate the total profit when 300 members are enrolled, we can compute the sum:

$\sum_{i=1}^{6} P'(x_i) \Delta x$

where $P'(x) = -0.0007x^3 + 0.39x^2 + 67.4x$, $\Delta x = 50$, and $x$ values are chosen at intervals of 50 from $x = 0$ up to $x = 300$.

Let's calculate $P'(x)$ at each $x_i$ where $x_i = 0, 50, 100, 150, 200, 250$, and then multiply each by $\Delta x = 50$ to find the total sum.

1. Calculate $P'(x_i)$ at each $x_i$:

   • $P'(0) = -0.0007(0)^3 + 0.39(0)^2 + 67.4(0) = 0$
   • $P'(50) = -0.0007(50)^3 + 0.39(50)^2 + 67.4(50)$
   • $P'(100) = -0.0007(100)^3 + 0.39(100)^2 + 67.4(100)$
   • $P'(150) = -0.0007(150)^3 + 0.39(150)^2 + 67.4(150)$
   • $P'(200) = -0.0007(200)^3 + 0.39(200)^2 + 67.4(200)$
   • $P'(250) = -0.0007(250)^3 + 0.39(250)^2 + 67.4(250)$

2. Multiply each result by $\Delta x = 50$ and sum them to approximate the total profit.

Let me compute these values.The approximate total profit when 300 members are enrolled is approximately $42,243.75 (in cents, this would be 4,224,375 cents).

Would you like a breakdown of these calculations, or do you have any other questions?

Here are some follow-up questions for a deeper understanding:

1. How does increasing or decreasing $\Delta x$ affect the accuracy of this approximation?
2. What is the significance of marginal profit in this context?
3. Can we interpret the marginal profit function $P'(x)$ for business insights?
4. How would the total profit change if we computed it up to 500 members?
5. How can integral calculus be used to get a more accurate result instead of using discrete sums?

Tip: When approximating a continuous function with discrete sums, smaller intervals ($\Delta x$) generally yield more accurate results.