Math Problem Statement

A manufacturer produces three products, A, B, and C. The profits for each unit of A, B, and C sold are $1, $2, and $3, respectively. Fixed costs are $16,000 per year, and the costs of producing each unit of A, B, and C are $4, $5, and $7, respectively. Next year, a total of 9000 units of all three products is to be produced and sold, and a total profit of $20,000 is to be realized. If total cost is to be $68,000, how many units of each of the products should be produced next year?

Solution

Let's break down the information given to form equations for this problem:

  1. Let xx, yy, and zz represent the units of products A, B, and C, respectively.

  2. Total units produced constraint:
    x+y+z=9000x + y + z = 9000

  3. Profit constraint:

    • Profit per unit for A, B, and C is $1, $2, and $3, respectively.
    • Total profit = $20,000. 1x+2y+3z=200001x + 2y + 3z = 20000

  4. Cost constraint:

    • Fixed costs are $16,000.
    • Production costs per unit for A, B, and C are $4, $5, and $7, respectively.
    • Total cost (including fixed cost) = $68,000. 4x+5y+7z+16000=680004x + 5y + 7z + 16000 = 68000 Simplifying, we get: 4x+5y+7z=520004x + 5y + 7z = 52000

We now have a system of three equations:

x + y + z = 9000 \\ x + 2y + 3z = 20000 \\ 4x + 5y + 7z = 52000 \end{cases}$$ Let's solve this system of equations to find $$x$$, $$y$$, and $$z$$.The solution to the system of equations is: - $$x = 3000$$ units of product A, - $$y = 1000$$ units of product B, - $$z = 5000$$ units of product C. Therefore, the manufacturer should produce: - 3000 units of product A, - 1000 units of product B, - 5000 units of product C. Would you like a detailed explanation of the steps taken, or do you have any questions? ### Related Questions 1. How would the solution change if the profit per unit of product A increased? 2. What if the production cost per unit of product C was reduced to $6? 3. How would the equations change if the fixed cost increased to $18,000? 4. What is the impact on production numbers if the profit goal was raised to $25,000? 5. Could these constraints lead to an optimal solution using linear programming? ### Tip In problems like this, setting up clear equations from the constraints is key to solving systems accurately.

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Math Problem Analysis

Mathematical Concepts

Algebra
System of Linear Equations

Formulas

Total units equation: x + y + z = 9000
Profit equation: 1x + 2y + 3z = 20000
Cost equation: 4x + 5y + 7z = 52000

Theorems

-

Suitable Grade Level

Grades 10-12

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