Difficult word problem below. I would appreciate your help.?

Question

A lamp has a cost function of C(x) = 2500 + 10x, where x is the number of units produced and C(x) is in dollars. The revenue function for these lamps is R(x) = 17x - 0.001x^2 . What is the cost, in dollars, of producing the number of lamps which maximizes the profit?

Here are some answers that myself and others have come up with..

$27,500
$32,500
$37,500
$42,500
$47,500
$52,500

Answer 1

Profit = revenue - cost

That's 17x - 0.001x^2 - 2500 - 10x

-0.001x^2 + 7x - 2500

This has a maximum when x = -b/(2a)

x = -7/(2 X -0.001) = 3500

The cost to make 3500 lamps is 2500 + 10(3500) = $37500

Answer 2

Profit = Revenue - Cost

Profit = 17x - .001x^2 - 2500 - 10x = 7x - .001x^2 - 2500

To maximize this you can do it either of 2 ways:
Recognize this is a parabola in form ax^2+bx+c and the x coordinated of the vertex is -b/2a and notice that since a is negative this is a maximum not a minimum

vertex = -7/2*-.001 = -7/-.002 = 3500

Or differentiate the equation and set it equal to zero and solve for x to find the maximum:

f(x) = -.001x^2 +7x - 2500
f'(x) = -.002x +7 = 0 => .002x = 7 => x = 7/.002 = 3500

C(x) = 2500 + 10x
C(3500) = 2500 + 10*3500 = 2500 + 35000 = 37500