Problem 75 - College Algebra and Calculus: An Applied Approach

Question

Consider a company that specializes in potting soil. Each bag of potting soil for seedlings requires 2 units of sand, 1 unit of loam, and 1 unit of peat moss. Each bag of potting soil for general potting requires 1 unit of sand, 2 units of loam, and 1 unit of peat moss. Each bag of potting soil for hardwood plants requires 2 units of sand, 2 units of loam, and 2 units of peat moss. Find the numbers of bags of the three types of potting soil that the company can produce with the given amounts of raw materials. 500 units of sand 500 units of loam 400 units of peat moss

Step-by-Step Solution

Answer

The company can produce 100 bags each of seedling, general potting, and hardwood plant soil.

Step 1: Formulate a system of linear equations

The system of equations can be written by taking the number of bags of seedling soil as x, general potting soil as y and hardwood soil as z. This gives us:\[2x + y + 2z = 500\] for sand, \[x + 2y + 2z = 500\] for loam and \[x + y + 2z = 400\] for peat moss.

Step 2: Solve the equations

Subtracting the third equation from the first gives: \[ x = 100\]. Substituting that into the other two equations gives: \[200 + 2y + 2z = 500\] and \[100 + y + 2z = 400\]. Subtracting these gives \[y = 100\] and substituting y and x into the third equation gives \[z = 100\].