- Sum all fixed monthly costs, which include permits (\(\$1,000\)), insurance (\(\$500\)), and base ingredient costs (\(\$200\)), totaling \(\$1,700\).
- Identify the variable cost per meal as \(\$3\), resulting in a variable expense component of \(3q\) for \(q\) meals.
- Construct the total expense function \(E(q)\) by adding the variable component to the sum of the fixed components.
- Conclude that the correct total expense equation is \(E(q) = 3q + 1,700\), which is option a.
- Final Answer: \(\boxed{a}\)
Explanation
- Analysis of Expenses
To determine the correct equation for the total monthly expenses \(E(q)\), we must differentiate between fixed and variable costs. Fixed costs are expenses that do not change regardless of the number of meals served, while variable costs depend directly on the quantity \(q\) of meals served. The provided data lists permit costs of \(\$1,000\), insurance costs of \(\$500\), and an additional monthly cost of \(\$200\) for ingredients, which, based on the problem description, is treated as a fixed monthly base cost. The variable cost is \(\$3\) per meal served.
- Calculating Fixed Monthly Expenses
The fixed monthly expenses are those that occur every month regardless of the food truck's activity level. Summing these costs gives:
\[\text{Total Fixed Costs} = \$1,000 (\text{permits}) + \$500 (\text{insurance}) + \$200 (\text{ingredients}) = \$1,700.\]
Thus, the base monthly cost the food truck incurs is \(\$1,700\).
- Determining Variable Monthly Expenses
The variable expense depends on the number of meals served, denoted by \(q\). Since each meal costs \(\$3\) to prepare, the total variable expense for \(q\) meals is simply \(3 \times q = 3q\). This cost scales linearly with the number of meals produced.
- Formulating the Total Expense Equation and Selecting the Correct Option
The total monthly expense function \(E(q)\) is the sum of the total fixed costs and the total variable costs:
\[E(q) = (\text{Total Variable Expenses}) + (\text{Total Fixed Expenses}) = 3q + 1,700.\]
Comparing this result to the given options, we find that it matches option a. Therefore, the correct equation is \(E(q) = 3q + 1,700\).
Examples
Understanding how to categorize costs into fixed and variable components is essential for any business owner. For example, a bakery owner needs to calculate their monthly expenses to set fair prices for their pastries. If they know their rent, electricity, and insurance (fixed costs) as well as the cost of flour, sugar, and eggs per pastry (variable costs), they can build a cost function like \(E(q) = \text{cost per item} \times q + \text{fixed base}\). This allows the owner to predict their total costs at different levels of production, helping them determine their break-even point and ensure long-term profitability.