The total cost of producing a type of tractor is given by C(x)=14000−70x+0.2x2, where x is the number of tractors produced. How many tractors should be produced to incur minimum cost?

Answer 1

C(x) = 14,000 -70x -+0.2x^2

find the derivative, set = 0, and solve for x to find the cost minimizing x value

C'(x) = -70 + 0.4x = 0

0.4x = 70

x = 70/0.4 = 700/4 = 175 tractors will minimize cost

C(175) = 14,000 -70x + 0.2x^2 = 14,000 -70(175) + 0.2(175)^2 = 14,000 -12,250 +30,625(.2)= 1,750+6125 = 7,875= minimum cost with 175 tractors produced

14,000 = fixed cost = total cost when zero tractors are produced or when 2(175) = 350 tractors are produced

-70x + 0.2x^2 = variable cost = -70(350)+0.2(350)^2 = 0 when 350 tractors are produced

the midpoint between x = 0 and x=350 is x=175 = cost minimizing level of output

graph the cost function and it's an upward opening parabola with axis of symmetry at x = 175. All x values to the left or right have higher cost. (175, 7875) is the vertex and minimum point of the parabola