- Identify fixed costs (\(\$4,000\) rent, \(\$500\) insurance) and variable components (\(\$20\) per car, \(\$300\) maintenance).
- Recognize the variable expenses equation \(V(q)\) must scale with the number of cars \(q\).
- Select the option that includes the rate per car (\(\$20q\)) and the stated variable cost constant (\(\$300\)).
- The correct equation is \(\boxed{V(q) = 20q + 300}\).
Explanation
- Analysis of Expenses and Variable Costs
To determine the correct variable expenses equation \(V(q)\), we must first distinguish between fixed and variable costs. Fixed monthly expenses remain the same regardless of how many cars are rented, while variable expenses change based on the number of cars, denoted by \(q\).
The provided information states:
- Fixed monthly expenses: Rent of \(\$4,000\) and insurance of \(\$500\), totaling \(\$4,500\). These costs are constant and do not depend on \(q\).
- Variable expenses: \(\$20\) per car rented and \(\$300\) for maintenance.
Although maintenance is often a fixed cost, the problem identifies it under the category of variable expenses. Since all provided options follow the linear form \(V(q) = 20q + k\), we infer that the term \(20q\) accounts for the cost per car, and the constant \(k\) includes the maintenance cost associated with the rental operation, which is \(\$300\).
- Formulating the Equation and Final Selection
Given the structure of the options and the description provided, we express the total variable expenses \(V(q)\) as:
\[V(q) = (\text{cost per car} \times q) + \text{maintenance cost}\]
\[V(q) = 20q + 300\]
This matches option (a). The other options add costs that are clearly categorized as fixed expenses (\(\$500\), \(\$4,000\), or a combination thereof), which do not belong in the equation for variable expenses. Thus, the equation \(V(q) = 20q + 300\) correctly models the variable expenses based on the rental service data.
Examples
Understanding how to distinguish between fixed and variable costs is essential for any business owner. For instance, if you operate a food truck, your daily permit fee is a fixed cost, regardless of how many meals you sell. Conversely, the cost of ingredients for each meal sold is a variable cost. By correctly separating these, you can calculate your break-even point—the exact number of meals you need to sell to cover all your expenses and start making a profit. This algebraic modeling helps in pricing strategy and operational planning.