Problem 108 - Beginning and Intermediate Algebra

Question

A manufacturer has fixed costs of \(\$ 1000\) to produce gizmos. Each gizmo costs \(\$ 5\) to make. The fixed cost to produce gadgets is \(\$ 750,\) and each gadget costs \(\$ 3\) to make. Write an expression for the cost to make \(y\) gadgets.

Step-by-Step Solution

Answer

The expression for the cost to make \(y\) gadgets is \(750 + 3y\).

Step 1 - Identify fixed costs

The fixed cost to produce gadgets is given as \(\$ 750\). This cost does not change regardless of the number of gadgets produced.

Step 2 - Identify variable cost per gadget

The variable cost to produce one gadget is given as \(\$ 3\). This means for each gadget produced, an additional \(\$ 3\) is incurred.

Step 3 - Set up the total cost expression

To find the total cost to produce \(y\) gadgets, combine the fixed costs and the variable costs. The total variable cost for \(y\) gadgets is \(3y\). Therefore, the total cost expression for making \(y\) gadgets will be the sum of the fixed cost and the total variable cost: \[ \text{Total Cost} = 750 + 3y \]

Key Concepts

Fixed Costs

Fixed costs are expenses that remain constant regardless of the production level. In our exercise, the fixed cost to produce gadgets is given as \( \$750 \). These costs do not change whether we produce one gadget or a thousand gadgets. Examples of fixed costs include:

• Rent
• Salaries of permanent staff
• Equipment depreciation

The primary characteristic of fixed costs is their invariability with production volume, providing companies with predictable expenses over certain periods even if they aren't producing anything.

Variable Costs

Variable costs, on the other hand, change in proportion to the level of production. For each gadget produced in our scenario, the cost is \( \$ 3 \). This means that if you produce more gadgets, the total variable costs increase. Examples of variable costs include:

• Raw materials
• Direct labor
• Packaging

To find the total variable cost of producing \( y \) gadgets, you multiply the variable cost per gadget by the number of gadgets produced. In this case, the variable cost expression is \( 3y \).

Total Cost Expression

The total cost expression combines both fixed and variable costs. This gives us a complete understanding of the overall production cost. The total cost for producing \( y \) gadgets can be expressed as:
\[ \text{Total Cost} = 750 + 3y \] In the equation:

• \( 750 \) represents the fixed costs
• \( 3y \) represents the total variable costs for \( y \) gadgets

By combining these two components, we capture the full expense needed for production. This formula highlights how both fixed and variable costs contribute to the overall financial resources required.