The cost of fuel for a boat is one half the cube of the speed on knots plus 216/hour. Find the most economical speed for the boat if it goes on a 500 nautical mile trip.

To find the most economical speed for the boat, we need to minimize the cost of fuel for a 500 nautical mile trip. Let's break down the problem step by step.

1. Let's denote the speed of the boat in knots as x.
2. The cost of fuel for the boat is given by the equation: cost = (1/2) * x^3 + 216/x.
3. We want to find the speed that minimizes the cost for a 500 nautical mile trip, which means we need to minimize the cost function with respect to x.
4. To do this, we can take the derivative of the cost function with respect to x and set it equal to zero.
cost' = 3/2 * x^2 - 216/x^2 = 0
5. Solve the equation 3/2 * x^2 - 216/x^2 = 0 for x.
Multiply through by 2x^2:
3x^4 - 432 = 0
Divide through by 3:
x^4 - 144 = 0
6. Rearrange the equation:
x^4 = 144
7. Take the fourth root of both sides:
x = ±∛144
8. Calculate the fourth root of 144:
x = ±3.3019
9. Since speed cannot be negative, we discard the negative value. Hence, x = 3.3019 knots.

Therefore, the most economical speed for the boat to travel on a 500 nautical mile trip is approximately 3.3019 knots.