The average cost is minimized when output is 31.6.

Average cost is minimized when marginal cost is equal to average cost. Given the total cost function, we first compute average cost, which is total cost divided by quantity, i.e.,

• average cost = (10y2+10000)/y=10y+10000/y(10y2+10000)/y=10y+10000/y

Marginal cost is the derivative of total cost with respect to quantity, i.e.,

• marginal cost = 20y20y

Equating marginal cost to average cost, we have:

• 10y+10000/y=20y10000/y=10y1000=y2y=31.6