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)=0.4x^2 - 104x + 13,586. How many cars must be made to minimize the unit cost? Do not round the answer

Question

12-10-2022
Mathematics

Answered

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)=0.4x^2 - 104x + 13,586. How many cars must be made to minimize the unit cost? Do not round the answer

Answer :

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sqdancefan sqdancefan · Answer 1 · 2022-10-12T13:32:51-04:00

Answer:

130 cars

Step-by-step explanation:

You want the value of x (the number of cars made) that minimizes the unit cost, given by C(x) = 0.4x² -104x +13586.

Vertex

The minimum cost will be found at the vertex of this quadratic cost function. For quadratic ax²+bx+c, the vertex is found at x=-b/(2a).

The cost function has a=0.4 and b=-104, so the number of cars that must be made to minimize the unit cost is ...

x = -b/(2a) = -(-104)/(2(0.4)) = 104/0.8

x = 130

130 cars must be made to minimize the unit cost.

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Additional comment

A graphing calculator can plot the cost function and show you the coordinates of the minimum cost. The attachment shows the minimum cost per car is $6826 when 130 cars are made.