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A pet salon has fixed monthly expenses of \$1,800 for rent, \$400 for insurance, and a line of credit payment of \$200. Initially, its variable expenses are given by $V(q)=5.00 q+100$, where $q$ is the number of pets styled. Due to an increase in the cost of styling supplies, an additional $\$1.00$ per pet is added to the variable expenses. What is the new equation for the total monthly expenses $E(q)$?


A$E(q)=6.00 q+2,500$
B$E(q)=6.00 q+2,550$
C$E(q)=6.00 q+2,300$
D$E(q)=5.00 q+2,550$

Verified Answer :

  • Sum the fixed monthly expenses to get \(\$2,400\).
  • Update the variable expenses by adding \(\$1.00\) per pet to the initial function \(\(5.00q + 100\)\), resulting in \(\(6.00q + 100\)\).
  • Add the total fixed expenses to the new variable expense function to obtain \(E(q) = 6.00q + 2,500\).
  • The correct option is \(\boxed{a. E(q) = 6.00q + 2,500}\).

Explanation

  1. Calculate Fixed Expenses

To determine the new total monthly expense function \(E(q)\), we first need to identify the components of the salon's costs. The fixed monthly expenses are the costs that remain constant regardless of the number of pets styled. These include:

  • Rent: \(\$1,800\)
  • Insurance: \(\$400\)
  • Credit payment: \(\$200\)

Summing these gives us the total fixed costs:
\[1,800 + 400 + 200 = \boxed{2,400}\]
So, the total fixed expense is \(\$2,400\).

  1. Determine New Variable Expenses

Next, we analyze the variable expenses. The initial variable expense function is \(V(q) = 5.00q + 100\), where \(q\) is the number of pets styled. The problem states that due to an increase in the cost of supplies, an additional \(\$1.00\) per pet is added to these expenses. We add this to the original variable expression:
\[V_{new}(q) = (5.00q + 100) + 1.00q = 6.00q + 100\]
This gives us the updated variable cost function.

  1. Calculate Total Monthly Expenses

Finally, the total monthly expense \(E(q)\) is the sum of the total fixed expenses and the new variable expenses:
\[E(q) = \text{Total Fixed Expenses} + V_{new}(q)\]
Substituting the values we calculated:
\[E(q) = 2,400 + (6.00q + 100)\]
Simplifying this expression:
\[E(q) = 6.00q + (2,400 + 100) = 6.00q + 2,500\]
Comparing this result to the given options, we find it corresponds to option a.

Examples

Understanding fixed versus variable expenses is essential for any business owner. For example, a bakery might have fixed costs like rent and kitchen equipment payments, while their variable costs include ingredients for each cake baked. By accurately calculating total expenses as a function of production, the owner can determine the price to charge per cake to ensure they cover all costs and make a profit. This mathematical modeling helps in making informed pricing decisions and maintaining financial stability.

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