Reference no: EM131604464

Question: In many applications, we want to maximize or minimize some quantity subject to a condition. Such constrained optimization problems are solved using Lagrange multipliers in multivariable calculus;

The quantity Q of an item which can be produced from quantities x and y of two raw materials is given by Q = 10xy at a cost of C = x + 2y thousand dollars. If the budget for raw materials is 10 thousand dollars, find the maximum production using the following steps.

(a) Graph x+2y = 10 in the first quadrant. On the same axes, graph Q = 10xy = 100, Q = 10xy = 200, and Q = 10xy = 300.

(b) Explain why the maximum production occurs at a point at which a production curve is tangent to the cost line C = 10.

(c) Using your answer to part (b) and implicit differentiation to find the slope of the curve, find the maximum production under this budget.