An orange grower has discovered a process for producing oranges that requires two inputs.The production function is Q  min2x1, x2, where x1 andx2 are the amounts of inputs 1 and 2 that he uses.The prices of these two inputs are w1  $5 and w2  $2,respectively.The minimum cost of producing 140 units is therefore

A) $980.
B) $630.
C) $1,400.
D) $280.
E) $700.

Question

Pablo Tarat · Accepted Answer

Final Answer : B, Explanation :  To find the minimum cost of producing 140 units, we need to use the equation for the cost minimization problem:Minimize C = w1x1 + w2x2subject to Q = F(x1, x2) = 140Using the production function Q = F(x1, x2) = 10 min{x1, 2x2}, we can rewrite it as x1 = 2x2 if x1 is the determining factor and x1 = x2/2 if x2 is the determining factor.If we substitute these expressions for x1 and x2 in the cost function C = w1x1 + w2x2, we get:C = w1(2x2) + w2x2/2 = 5(2x2) + 2(x2/2) = 11x2Now, we can minimize the cost function by differentiating it with respect to x2 and setting the derivative equal to zero:dC/dx2 = 11 = 0x2 = 0This is clearly not a valid solution, so we can try other methods. For example, we can use the constraint Q = F(x1, x2) = 140 to eliminate one of the variables. Since we want to minimize cost, we can use the equation x1 = 2x2 to eliminate x1 in favor of x2, since x2 is cheaper. Then we have:Q = 10 min{x1, 2x2} = 10x1 = 20x2140 = 20x2x2 = 7Therefore, x1 = 2x2 = 14, and the minimum cost is C = w1x1 + w2x2 = 5(14) + 2(7) = 98. So the best choice is B.

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