The maximum profit potential for the manufacturer is approximately $32,176.78.
Find the derivative of π(x):
π'(x) = 9√(900-x) - (3x * (1/2√(900-x))) = (18√(900-x) - 3x)/2√(900-x)
Set the derivative to zero and solve for x (critical points):
(18√(900-x) - 3x)/2√(900-x) = 0
18√(900-x) - 3x = 0
3x = 18√(900-x)
x^2 = 6√(900-x) * x
x^2 - 6√(900-x) * x = 0
This equation can be factored as:
x(x - 6√(900 - x)) = 0
Therefore, x = 0 or x = 6√(900 - x).
Check the critical points and endpoints (0 and 900) for maximum profit:
π(0) = 3 * 0 * √(900-0) = 0
π(900) = 3 * 900 * √(900-900) = 0
π(6√(900 - 6√(900))) = 3 * 6√(900 - 6√(900)) * √(900 - (6√(900))) ≈ 32,176.78 (maximum profit)
So the answer is B. $32,176.78.