Calculus problem solving- min ave cost & parabola?

Question

A. If the cost of producing "x" units of a certain product is given by C= 10000 + 5x + x^2/9.

Find the value of x that gives the minimum average cost per unit. Please note that the tangent is horizontal at a minimum point.

(the tangent is horizontal so slope is 0?)

B. Consider a pt in the first quadrant and on the parabola y=4-x^2. If we draw a tangent at this point, the tangent , x axis and y axis form a triangle. Find the coordinates of the point on the curve so that the area of this triangle is 6.5 units.

(I got 1/13, Im not sure though. And they say the other answer is 2705/676. I dont know how to get it)

Answer 1

A. Given the cost of producing x units:
C = 10000 + 5x + x²/9.
Find x such that the average cost per unit is minimized.

The average cost per unit is V = C/x.
V = 10000/x + 5 + x/9

Take the derivative to find the critical points.

dV/dx = -10,000/x² + 1/9 = 0
10,000/x² = 1/9
90,000 = x²
x = ±300
x = 300 since the negative solution is rejected

Take the second derivative to find the nature of the critical point.

d²V/dx² = 20,000/x³ > 0
This implies a relative minimum which is what we want.

So the answer is x = 300 units results in the lowest average cost per unit of production.