Exam 14: Functions of Two or More Variables

The demand functions for two products are given by ​ ​ Where p₁ and p₂ are in dollars and q₁ and q₂ are numbers of units. Find a pair of prices p₁ and p₂ such that the demands for the products will be equal. Round your answers to the nearest cent. ​

Suppose that the production function for a product is , where x represents the number of work hours per month and y is the number of available machines. Determine the marginal productivity of x. ​

Suppose that the number of tons of an agricultural product produced is given by , where x is the number of hours of labor and y is the number of acres of the crop. Find the number of tons produced when and . Round your answer to the nearest number of tons. ​

A firm has two plants, X and Y. Suppose that the cost of producing x units at plant X is dollars and the cost of producing y units of the same product at plant Y is given by dollars. If the firm has an order for 3,000 units, how many should it produce at each plant to fill this order and minimize the cost of production? ​

Find the minimum value of subject to the condition . ​

Suppose that the profit from the sale of Kandy and Kreams is given by dollars, where x is the number of pounds of Kandy and y is the number of pounds of Kreams. How many pounds of Kandy and Kreams must be sold to maximize profit? What is the maximum profit? Round your pounds answers to the nearest whole number. Round your max profit answer to the nearest cent. ​

Find the maximum value of subject to , , . Round your answer to the nearest integer. ​

Give the domain of the function . ​

The material for the base of an open box costs 1.5 times as much per unit area as the material for constructing the sides. Use Lagrange multipliers to find the dimensions of the box of largest volume that can be made for a fixed cost of 200.00. Round your answers to two decimal places. (Maximize subject to .) ​

Suppose that the utility function for two products is given by , and the budget constraint is . Find the values of x and y that maximize utility. ​

Suppose that the output Q (in units) of a certain company is , where K is the capital expenditures in thousands of dollars and L is the number of labor hours. Find and when capital expenditures are $45,000 and the labor hours total 5,400. Round your answers to two decimal places. ​

If , find . ​

The demand functions for q_A and q_B units of two related products, A and B, are given. ​ ​ Where p_A and p_B are in dollars. Find the marginal demand of q_A with respect to p_B and the marginal demand of q_B with respect to p_A. Are the two goods competitive or complementary? ​

Suppose that the number of units of a good produced z is given by , where x is the number of machines working properly and y is the average number of work hours per machine. Find the production for a week in which 9 machines are working properly and the average number of work hours per machine is 35. Round your answer to the nearest whole number. ​

If , find . ​

Give the domain of the function . ​

If the joint cost function for two products is (in dollars). Find the marginal cost with respect to y.

Suppose that tons is the production function for a product with x units of one input and y units of a second input. Find the values of x and y that will maximize production. ​

If , find .

Find x and y such that subject to the constraint attains a minimum value. ​