Find the number of units that should be produced at each location to minimize the cost.

Question

A manufacturer has an order for 7000 units of fine paper that can be produced at two locations. Let x1 and x2 be the number of units produced at the two locations. The cost function is modeled by:

C=0.1x1^2+25x1+0.05x2^2+48x2

Find the number of units that should be produced at each location to minimize the cost.

x1= ______ units

x2= ______ units

Answer 1

The total number of units is 7000 so x₁ + x₂ = 7000 or x₂ = 7000 - x₁

Replacing x₂ with "7000 - x₁" in the cost equation gives us:

C(x₁) = 0.1x₁² + 25x₁ + 0.05(7000 - x₁)² + 48(7000 - x₁)

C(x₁) = 0.1x₁² + 25x₁ + 0.05(49E6 - 14000x₁ + x₁²) + 336000 - 48x₁

C(x₁) = 0.1x₁² + 25x₁ + 2450000 - 700x₁ + 0,05x₁² + 336000 - 48x₁

C(x₁) = 0.15x₁² - 723x₁ + 2786000

To minimize, take the derivative and set it equal to zero

C'(x₁) = 0.30x₁ - 723

0 = 0.30x₁ - 723

0.30x₁ = 723

x₁ = 723/0.30 = 2410 units

Since x₂ = 7000 - x₁ then x₂ = 7000 - 2410 or x₂ = 4590 units