Questions: You are placing mulch in your yard, and you find that pine chips cost 3 per bag, while oak chips cost 5 per bag. You want to minimize total cost. Let x be the number of bags of pine chips and y be the number of bags of oak chips. Define the objective function for this problem. c=

You are placing mulch in your yard, and you find that pine chips cost 3 per bag, while oak chips cost 5 per bag. You want to minimize total cost. Let x be the number of bags of pine chips and y be the number of bags of oak chips. Define the objective function for this problem.

c=

Transcript text: You are placing mulch in your yard, and you find that pine chips cost $\$ 3$ per bag, while oak chips cost $\$ 5$ per bag. You want to minimize total cost. Let $x$ be the number of bags of pine chips and $y$ be the number of bags of oak chips. Define the objective function for this problem. \[ c= \]

Solution

Solution Steps

To define the objective function for minimizing the total cost of purchasing pine and oak chips, we need to express the total cost in terms of the number of bags of each type of chips. The cost per bag of pine chips is $3, and the cost per bag of oak chips is $5. Therefore, the total cost function can be written as:

\[ c = 3x + 5y \]

where $ x $ is the number of bags of pine chips and $ y $ is the number of bags of oak chips.

Step 1: Define the Variables

Let $ x $ be the number of bags of pine chips and $ y $ be the number of bags of oak chips.

Step 2: Write the Objective Function

The total cost $ c $ for purchasing the bags of chips can be expressed as: \[ c = 3x + 5y \]

Step 3: Calculate the Total Cost for Given Values

Substituting $ x = 10 $ and $ y = 5 $ into the objective function: \[ c = 3(10) + 5(5) = 30 + 25 = 55 \]