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Delivery Service Company Expenses



Imagine you own a small delivery service company. Your monthly expenses include the following:



Fixed expenses:

  • Rent: \$1,800
  • Insurance: \$500
  • Equipment lease: \$300



Variable expenses:

  • Fuel cost per delivery: \$2
  • Maintenance per delivery: \$0.50
  • Driver's wage per delivery: \$5



You must create an equation to model the total monthly expenses $E(q)$ based on the number of deliveries $q$.



Imagine that the delivery service company's rent increases to \$2,000 from \$1,800.

The updated equation to model the total monthly expenses $E(q)$ is:

$E(q) = 7.50q +$ [ ]


A2,500
B2,600
C2,700
D2,800

Verified Answer :

  • Identify the total fixed expenses: Rent (\$2,000) + Insurance (\$500) + Equipment lease (\$300) = \$2,800.
  • Calculate the total variable expense per delivery: Fuel (\$2) + Maintenance (\$0.50) + Driver's wage (\$5) = \$7.50 per delivery.
  • Construct the total cost equation: \(E(q) = 7.50q + 2,800\).
  • The missing constant \(\square\) is \(\boxed{2,800}\).

Explanation

  1. Problem Analysis and Data Categorization

We are analyzing the monthly expenses of a delivery service company to model its total cost \(E(q)\) as a function of the number of deliveries \(q\). The costs are categorized into fixed expenses (costs that do not depend on the number of deliveries) and variable expenses (costs that accrue for each delivery).

Initially, the fixed expenses were given as Rent = \$1,800, Insurance = \$500, and Equipment lease = \$300. However, the problem specifies an update where the rent increases to \$2,000.

The variable expenses per delivery are Fuel = \$2, Maintenance = \$0.50, and Driver's wage = \$5. The total variable cost per delivery is the sum of these individual costs.

  1. Calculating Total Expenses

To find the total monthly expense equation \(E(q)\), we calculate the total fixed expenses and the total variable cost per delivery.

Total fixed expenses:
\[\text{Total Fixed} = \text{Updated Rent} + \text{Insurance} + \text{Equipment lease}\]
\[\text{Total Fixed} = 2,000 + 500 + 300 = 2,800\]

Total variable cost per delivery:
\[\text{Total Variable per } q = \text{Fuel} + \text{Maintenance} + \text{Driver's wage}\]
\[\text{Total Variable per } q = 2 + 0.50 + 5 = 7.50\]

  1. Formulating the Equation and Identifying the Constant

The total monthly expense equation is given by:
\[E(q) = (\text{Total Variable per } q) \times q + \text{Total Fixed}\]
Substituting the values calculated:
\[E(q) = 7.50q + 2,800\]
Comparing this to the provided equation \(E(q) = 7.50q + \square\), we identify that the missing constant \(\square\) corresponds to the total fixed expenses, which is \$2,800. Comparing this with the options provided (2,500, 2,600, 2,700, 2,800), the correct value is 2,800.

Examples

In business management, accurately modeling expenses is crucial for pricing services and ensuring profitability. For example, if a ride-sharing driver understands their fixed costs (like car insurance and lease) and their variable costs (like fuel and wear-and-tear per mile), they can calculate exactly how much each trip must earn to cover these costs and contribute to their profit. Using linear equations \(E(q) = mq + c\) allows business owners to quickly forecast their budget at different levels of operational activity.

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