How many tractors?

Question

The total cost of producing a type of tractor is given by C(x)=25000−30x+0.05x^2, where x is the number of tractors produced. How many tractors should be produced to incur minimum cost?

Answer 1

take the derivative and set equal to zero; C' = -30 + x/10=0 . solve for x = 300

Produce 300 and cost = 25,000-30(300) + (1/20)(300)^2 = 25,000

Produce 0 and cost = 25,000.

Produce 1 and cost = 25K-30 + 1/20 = $24,970.05

Produce 2 and cost = 25K-60 + 4/20 = $24,940.20

Produce 10 and cost = 25K-300+ 100/20 = $24705

Produce 20 C = 25K -600 + 20 = $24,420

x=60 C = 25K -2K + 180 = $23,180

x= 100 C = 25K -3K +K/2 = $22,500

x=200 C = 25K -6K +2K = $21,000

x=300 C = 25K -9K + 4.5K = $20,500 minimum cost

x=400 C = 25K- 12K + 8K = $21,000

vertex of the upward opening parabola is the minimum point, (x,C) = (300,20,500)

Answer 2

Find h

This is a polynomial so rewrite it as

.05x²-30x+25000 and find the vertex

(because it is a positive polynomial, there will be a minimum point as the vertex)

x=h, h= -b/2a

so h=30/2(.05)=300, x=300

300 tractors