A photography studio has fixed monthly expenses of \$2,200 for rent and \$600 for insurance. Its variable expenses are given by $V(q)=15 q+200$, where $q$ is the number of photo sessions conducted. Which equation models the total monthly expenses $E(q)$ based on the number of photo sessions $q$?

A$E(q)=15 q+2,400$
B$E(q)=15 q+3,000$
C$E(q)=15 q+200$
D$E(q)=15 q+2,800$

To find the total monthly expenses $E(q)$, we must sum the fixed costs and the variable costs associated with the number of photo sessions, denoted by $q$. The fixed monthly costs consist of the rent and the insurance. The variable expenses are described by the function $V(q) = 15q + 200$.

Calculating Total Expenses

The fixed monthly expenses are calculated by adding the cost of rent and the cost of insurance:
\[ \text{Fixed Expenses} = \$2,200 + \$600 = \$2,800 \]
The total monthly expenses $E(q)$ are the sum of these fixed expenses and the variable expense function $V(q)$:
\[ E(q) = \text{Fixed Expenses} + V(q) \]
\[ E(q) = 2,800 + (15q + 200) \]

Simplifying and Selecting the Final Expression

Now, we simplify the expression for $E(q)$ by combining the constant terms:
\[ E(q) = 15q + (2,800 + 200) \]
\[ E(q) = 15q + 3,000 \]
Comparing this result to the given options, we find that it matches option (b).

Examples

This type of cost modeling is essential in business management, where understanding fixed vs. variable costs allows owners to predict their monthly budget based on business volume. For instance, a small business owner can use this linear function to forecast profitability or determine how many clients are needed to cover all expenses, ensuring the company remains financially sustainable.