Problem 109 - 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 total cost to make \(x\) gizmos and \(y\) gadgets.

Step-by-Step Solution

Answer

Total cost = 1750 + 5x + 3y

Step 1: Identify Fixed Costs

The fixed cost for making gizmos is \(1000 and for gadgets is \)750.

Step 2: Identify Variable Costs

The cost to make each gizmo is \(5 and to make each gadget is \)3.

Step 3: Formulate Expression for Total Cost of Gizmos

The total cost for producing x gizmos is the fixed cost for gizmos plus the variable cost times the number of gizmos: \[ 1000 + 5x \]

Step 4: Formulate Expression for Total Cost of Gadgets

The total cost for producing y gadgets is the fixed cost for gadgets plus the variable cost times the number of gadgets: \[ 750 + 3y \]

Step 5: Combine Expressions

Sum the total costs for gizmos and gadgets to get the overall total cost: \[ 1000 + 5x + 750 + 3y = 1750 + 5x + 3y \]

Key Concepts

Fixed Costs

Fixed costs refer to expenses that do not change with the level of production. Whether you produce one unit or one thousand units, fixed costs remain constant. Examples include rent, salaries, and insurance. In our exercise, the manufacturer has fixed costs of \(\$ 1000\) for gizmos and \(\$ 750\) for gadgets. These costs must be paid regardless of how many gizmos or gadgets are produced.

Variable Costs

Variable costs, unlike fixed costs, change with the level of production. These costs increase as you produce more units and decrease as you produce fewer. Examples include raw materials, electricity used in production, and labor directly involved in manufacturing. In our example, the variable cost to produce each gizmo is \(\$ 5\) and for each gadget, it is \(\$ 3\). Therefore, if you produce \(x\) gizmos, the total variable cost for gizmos is \(5x\). Similarly, if you produce \(y\) gadgets, the total variable cost for gadgets is \(3y\).

Total Cost Formula

The total cost is the sum of fixed and variable costs. It can be represented as an equation that includes the fixed costs, variable costs per unit, and the quantity of each product produced.
For gizmos, the total cost (T) is given by: \[ T_{\text{gizmos}} = \text{Fixed Cost}_{\text{gizmos}} + (\text{Variable Cost}_{\text{per gizmo}} \times \text{Quantity}_{\text{gizmos}}) = 1000 + 5x \].
For gadgets, the total cost (T) is given by: \[ T_{\text{gadgets}} = \text{Fixed Cost}_{\text{gadgets}} + (\text{Variable Cost}_{\text{per gadget}} \times \text{Quantity}_{\text{gadgets}}) = 750 + 3y \].
Combining both, the overall total cost (T) of producing \(x\) gizmos and \(y\) gadgets is: \[ T_{\text{total}} = 1000 + 5x + 750 + 3y \. \] Simplifying, we get: \[ T_{\text{total}} = 1750 + 5x + 3y \].