To approximate the total profit when 300 members are enrolled using the given marginal profit function, we can compute the summation as described:

$\text{Given: } P'(x) = -0.0005x^3 + 0.22x^2 + 53.9x,\ \Delta x = 50,\ x \text{ ranges from } 0 \text{ to } 300.$

We divide the interval $[0, 300]$ into 6 subintervals, each of width $\Delta x = 50$. For $x_i$, we take the endpoints of each subinterval:

$x_1 = 50,\ x_2 = 100,\ x_3 = 150,\ x_4 = 200,\ x_5 = 250,\ x_6 = 300.$

The sum is given by:

$\text{Total Profit} \approx \sum_{i=1}^6 P'(x_i) \Delta x.$

Step-by-Step Computation:

1. Compute $P'(x_i)$ for each $x_i$: $P'(x) = -0.0005x^3 + 0.22x^2 + 53.9x.$

   • For $x_1 = 50$: $P'(50) = -0.0005(50)^3 + 0.22(50)^2 + 53.9(50) = -0.0005(125000) + 0.22(2500) + 2695 = -62.5 + 550 + 2695 = 3182.5.$

   • For $x_2 = 100$: $P'(100) = -0.0005(100)^3 + 0.22(100)^2 + 53.9(100) = -0.0005(1000000) + 0.22(10000) + 5390 = -500 + 2200 + 5390 = 7090.$

   • For $x_3 = 150$: $P'(150) = -0.0005(150)^3 + 0.22(150)^2 + 53.9(150) = -0.0005(3375000) + 0.22(22500) + 8085 = -1687.5 + 4950 + 8085 = 11347.5.$

   • For $x_4 = 200$: $P'(200) = -0.0005(200)^3 + 0.22(200)^2 + 53.9(200) = -0.0005(8000000) + 0.22(40000) + 10780 = -4000 + 8800 + 10780 = 15580.$

   • For $x_5 = 250$: $P'(250) = -0.0005(250)^3 + 0.22(250)^2 + 53.9(250) = -0.0005(15625000) + 0.22(62500) + 13475 = -7812.5 + 13750 + 13475 = 19412.5.$

   • For $x_6 = 300$: $P'(300) = -0.0005(300)^3 + 0.22(300)^2 + 53.9(300) = -0.0005(27000000) + 0.22(90000) + 16170 = -13500 + 19800 + 16170 = 22470.$

2. Multiply each $P'(x_i)$ by $\Delta x = 50$: $P'(50) \Delta x = 3182.5 \times 50 = 159125,$ $P'(100) \Delta x = 7090 \times 50 = 354500,$ $P'(150) \Delta x = 11347.5 \times 50 = 567375,$ $P'(200) \Delta x = 15580 \times 50 = 779000,$ $P'(250) \Delta x = 19412.5 \times 50 = 970625,$ $P'(300) \Delta x = 22470 \times 50 = 1123500.$

3. Sum these values: $\text{Total Profit} = 159125 + 354500 + 567375 + 779000 + 970625 + 1123500 = 3858125 \, \text{cents}.$

Convert to Dollars:

$\text{Total Profit} = 3858125 \, \text{cents} = 38581.25 \, \text{dollars}.$

Final Answer:

The total profit when 300 members are enrolled is approximately $38,581.25.

Would you like further clarifications or detailed steps for any part of this calculation?

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Relative Questions:

1. How does the marginal profit function relate to total profit in this problem?
2. Why is the interval $[0, 300]$ divided into 6 subintervals?
3. What would happen to the accuracy if smaller intervals were chosen?
4. How can this summation method approximate an integral?
5. Can we calculate the exact total profit by integrating $P'(x)$?

Tip:

When approximating sums, smaller intervals ($\Delta x$) improve accuracy because they better capture the curve of the function.