A manufacturer has an order for 1100 units of fine paper that can be produced at two locations. Let $x _ { 1 }$ and $x _ { 2 }$ be the numbers of units produced at the two plants. Find the number of units that should be produced at each plant to minimize the cost if the cost function is given by $C = 0.2 x _ { 1 } ^ { 2 } + 25 x _ { 1 } + 0.05 x _ { 2 } ^ { 2 } + 12 x _ { 2 }$ .
A) $x _ { 1 } = 388$ units and $x _ { 2 } = 906$ units
B) $x _ { 1 } = 906$ units and $x _ { 2 } = 194$ units
C) $x _ { 1 } = 194$ units and $x _ { 2 } = 906$ units
D) $x _ { 1 } = 194$ units and $x _ { 2 } = 1812$ units
E) $x _ { 1 } = 1812$ units and $x _ { 2 } = 388$ units

Question

Quizplus · Accepted Answer

To minimize the cost, we need to set up the problem as a constrained optimization problem. The constraint is $x_1 + x_2 = 1100$, since the total production must equal 1100 units. The objective function to minimize is $C = 0.2x_1^2 + 25x_1 + 0.05x_2^2 + 12x_2$. To find the minimum cost, we would take the partial derivatives of C with respect to $x_1$ and $x_2$, set them equal to zero, and solve the system of equations along with the constraint. Without going through all the algebraic steps here, the correct answer reflects the optimal production levels at each plant to minimize the cost while meeting the production requirement.

A Manufacturer Has an Order for 1100 Units of Fine x1x _ { 1 }x1​

A Manufacturer Has an Order for 1100 Units of Fine $x _ { 1 }$