Final answer:
To find the quantity of PlayStation 4's manufactured each hour that minimizes the average cost, differentiate the cost function and solve for where the derivative equals zero. This value of x gives the optimal quantity. To find the corresponding minimum average cost, plug this value of x into the average cost function.
Step-by-step explanation:
To minimize the average cost for manufacturing PlayStation 4's, we first need to find the cost function C(x) which is given as C(x) = 800,000 + 340x + 0.0005x². The average cost AC(x) can be found by dividing the total cost C(x) by the number of units x, resulting in AC(x) = C(x)/x. However, to find the minimum average cost, we can differentiate the total cost C(x) and find the critical points to determine where the cost is minimized.
To do this, we differentiate C(x) with respect to x to find C'(x), which is C'(x) = 340 + 0.001x. Setting C'(x) equal to zero and solving for x gives us the number of PlayStation 4's that should be manufactured to minimize the average cost. Once we have the value of x, we can plug it back into the average cost function AC(x) to determine the minimum average cost.