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A fitness center has fixed monthly expenses of \$2,500 for rent, \$300 for equipment maintenance, and a line of credit payment of $\$ 400$. Its variable expenses are given by $V(q)=10 q+250$, where $q$ is the number of memberships sold. What is the correct equation for the total monthly expenses $E(q)$?


A$E(q)=10 q+3,150$
B$E(q)=10 q+2,750$
C$E(q)=10 q+3,450$
D$E(q)=10 q+3,050$

Verified Answer :

  • Sum all fixed costs: \(\$2,500 + \$300 + \$400 = \$3,200\).
  • Add the fixed costs to the variable expense function: \(E(q) = 3,200 + (10q + 250)\).
  • Simplify the expression by combining constants: \(E(q) = 10q + 3,450\).
  • Match the result with the given options to find \(\boxed{E(q) = 10q + 3,450}\).

Explanation

  1. Analysis of Expenses

The total monthly expenses \(E(q)\) for the fitness center are composed of the sum of its fixed monthly costs and its variable costs. The fixed costs remain constant regardless of the number of memberships sold, while the variable costs depend directly on the number of memberships \(q\). The problem provides the fixed expenses: rent \(\$2,500\), equipment maintenance \(\$300\), and a line of credit payment \(\$400\). The variable expenses are given by the function \(V(q) = 10q + 250\).

  1. Calculating the Total Expense Equation

First, we calculate the total fixed expenses \(F\):
\[F = 2500 + 300 + 400 = 3200.\]
Next, we define the total monthly expense equation \(E(q)\) as the sum of total fixed expenses and the variable expense function:
\[E(q) = F + V(q)\]
Substituting the known values:
\[E(q) = 3200 + (10q + 250)\]
Combining the constant terms:
\[E(q) = 10q + (3200 + 250) = 10q + 3450.\]

  1. Final Result Selection

Comparing our derived equation \(E(q) = 10q + 3,450\) with the given options:

A\(E(q) = 10q + 3,150\)
B\(E(q) = 10q + 2,750\)
C\(E(q) = 10q + 3,450\)
D\(E(q) = 10q + 3,050\)

The derived equation matches option c. Therefore, the correct equation for the total monthly expenses is \(\boxed{E(q) = 10q + 3,450}\).

Examples

Understanding total cost structures is essential for business management. For instance, if you operate a subscription service, distinguishing between your overhead (rent, electricity) and per-user costs (data usage, customer support) allows you to calculate your break-even point and set membership prices appropriately. In this scenario, you accurately mapped your fixed and variable outlays to determine exactly how your total monthly expenditure scales as you acquire more customers.

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