A factory can produce two products, x and y, with a profit P approximated by P = 12x + 15y. The production of y can exceed x by no more than 200 units. Moreover, production levels are limited by the formula x + 2y ≤ 1000. What production levels yield maximum profit?
Question
A factory can produce two products, x and y, with a profit P approximated by P = 12x + 15y. The production of y can exceed x by no more than 200 units. Moreover, production levels are limited by the formula x + 2y ≤ 1000. What production levels yield maximum profit?
Solution
To find the production levels that yield maximum profit, we need to solve this as a linear programming problem. Here are the steps:
-
Identify the objective function: The profit P = 12x + 15y is the objective function that we want to maximize.
-
Identify the constraints: The constraints are given by the inequalities y ≤ x + 200 and x + 2y ≤ 1000.
-
Graph the feasible region: Plot the constraints on a graph. The feasible region is the area that satisfies all the constraints. In this case, it's the area below the lines y = x + 200 and x + 2y = 1000.
-
Identify the corner points: The corner points of the feasible region are where the maximum or minimum values of the objective function occur. In this case, the corner points are (0,0), (0,500), (333.33, 333.33), and (400, 300).
-
Evaluate the objective function at the corner points: Calculate the profit P at each of the corner points:
P(0,0) = 12(0) + 15(0) = 0 P(0,500) = 12(0) + 15(500) = 7500 P(333.33, 333.33) = 12(333.33) + 15(333.33) = 7999.92 P(400, 300) = 12(400) + 15(300) = 8100
-
Determine the maximum value: The maximum profit is $8100, which occurs at the production levels x = 400 and y = 300.
Similar Questions
A consumer has a budget of $3000 in a given period, and wishes to buy two goods, X and Y, so as to maximise his utility. The price of good X is $5 and the price of good Y is $2, and his MRS is given by the formula 2Y/X. How many units of good X will he buy in that period?
. Given the following data for a perfectly competitive firm, what is the amount of profit the firm will make atthe profit maximizing output? AR = $100 and TC = 1,000 + 125Q - .5Q2 where Q = units produced permonthA) none – the firm would shut downB) $1,312.50C) $2,548.63D) –$1,425.86E) –$2,351.27
Suppose that a firm produces 200,000 units a year and sells them all for $10 each. The explicit costs of production are $1,500,000 and the implicit costs of production are $300,000. The firm earns an accounting profit of ________ and an economic profit of ________.Multiple Choice$500,000; $200,000$500,000; $1,700,000$1,700,000; $200,000$1,700,000; $500,000
Our company sells a product for $120 per unit. Variable costs are $90 per unit and fixed costs are $4,000. The company expects to sell 200 units this year. How many units must we sell to earn a profit of $2,000?Group of answer choices200250300340
A company manufactures three products. The following information is obtained in respect of nextmonth’s budgeted production.product X product Y product Zcontribution per unit $7 $6 $8contribution per kilo $3 $4 $6kilos of material requiredfor production400 600 1000The company has been advised that only 1800 kilos of material will be available for productionnext month.What is the maximum contribution the company can earn?A $9000 B $9600 C $13 000 D $13 200
Upgrade your grade with Knowee
Get personalized homework help. Review tough concepts in more detail, or go deeper into your topic by exploring other relevant questions.