Question

a lumber mill can turn out 600 units of plywood each week. to meet the needs of its regular customers, the mill must produce 150 units of lumber and 225 units of plywood. if the profit for each unit of lumber is $30 and for each unit of plywood is $45, how many units of each type wood product should the mill produce to maximize profit?

Answer 1

Step-by-step explanation:

If x = units of lumber and y = units of plywood, then:

x + y ≤ 600

x ≥ 150

y ≥ 225

Graph the inequalities. The lines intersect at three points:

(150, 225)

(150, 450)

(375, 225)

Evaluate the profit at each intersection:

P = 30x + 45y

P = 30(150) + 45(225) = 14,625

P = 30(150) + 45(450) = 24,750

P = 30(375) + 45(225) = 21,375

The highest profit is at 150 units of lumber and 450 units of plywood.

Answer 2

The question asked involves using linear programming to maximize profits from the production of lumber and plywood. The optimization can be set up with a profit function and constraints for production capacities. Since some constraints are missing, the problem is not fully solvable with the given information.

Step-by-step explanation:

The student has asked for help determining the optimal production of lumber and plywood at a lumber mill to maximize profits, given a specific production capacity and profit per unit for each product. To solve the problem, we can use linear programming. However, the student's question seems incomplete, as the constraints are not fully specified regarding the total production capacity for lumber. We only know the mill can produce 600 units of plywood per week, and must meet a regular customer need of 150 units of lumber and 225 units of plywood.

Nevertheless, the profit optimization problem can be generally set up as follows:

• Let x be the number of units of lumber produced.
• Let y be the number of units of plywood produced.
• The profit function to be maximized is P = 30x + 45y.

Assuming the mill can produce x units of lumber and y units of plywood, we would set up inequality constraints based on the given production requirements:

• x ≥ 150 (minimum lumber production)
• y ≥ 225 (minimum plywood production)
• x + y ≤ production capacity (if given)

Without the total production capacity for both lumber and plywood, we cannot complete the optimization. Once all constraints are known, the mill would use methods such as the Simplex algorithm to find the values of x and y that maximize the profit function P within those constraints.