An aircraft factory manufactures airplane engines. The unit cost C (the cost in dollars to make each airplane engine) depends on the number of engines made. If x engines are made, then the unit cost is given by the function C(x)=0.5x^2-330x+65,738. How many engines must be made to minimize the unit cost? Do not round your answer

If the unit cost is given by the function C(x)=0.5x^2-330x+65,738 where x is the amount of engines made while C(x) is the unit cost. In order to know the amount of engine needed to minimize the cost;

we will first find dC/dx and equate the resulting function to zero to get x.

dC/dx = 2(0.5)x - 330

dC/dx = x - 330

since dC/dx = 0

x - 330 = 0

x = 330 (critical point)

Then find the second derivative to check if it is greater than zero.

d²C/dx² = 1

Since d²C/dx² > 0, then the cost will be at the minimum when x = 330

Substitute x = 330 into the modeled function;

C(330)=0.5(330)^2-330(330)+65,738.

C(330)= 54,450-108,900+65,738.

C(330)= -54,450+65,738.

C(330)= 11,288

Hence the amount of engines that must be made to minimize the unit cost is 330 engines and the unit cost at the minimum price is 11,288