Question

A vehicle factory manufactures cars. The unit cost C (the cost in dollars to make each car) depends on the number of cars made. If x cars are made, then the unit cost is given by the function C(x) = 0.9x^2 -234x + 23,194 . How many cars must be made to minimize the unit cost?

Do not round your answer.

Answer 1

130 cars.

Explanation:

The cost function is given by:

C(x) = 0.9x^2 -234x + 23,194; where x is the input and C is the total cost of production.

To find the minimum the unit cost, there must be a certain number of cars which have to be produced. To find that, take the first derivative of C(x) with respect to x:

C'(x) = 2(0.9x) - 234 = 1.8x - 234.

To minimize the cost, put C'(x) = 0. Therefore:

1.8x - 234 = 0.

Solving for x gives:

1.8x = 234.

x = 234/1.8.

x = 130 units of cars.

To check whether the number of cars are minimum, the second derivative of C(x) with respect to x:

C''(x) = 1.8. Since 1.8 > 0, this shows that x = 130 is the minimum value.

Therefore, the cars to be made to minimize the unit cost = 130 cars!!!