Answer:
25 cupcakes
Explanation:
Simplifying the problem further with the specific terminology:
- We're essentially trying to find how many cupcakes must be sold in order to "break-even" (i.e., we want the revenue earned from the cupcakes to equal the cost of producing the cupcakes).
- We can do this using the linear revenue and cost functions.
General equation for the revenue function and revenue function in context:
The revenue function is the product of the price (p) and quantity (q) of an item:
Revenue = price * quantity
Thus, the general equation of the revenue function is given by:
R(q) = pq, where
- R is the revenue per q items sold,
- p is the price of the item,
- and q is the quantity of the item.
Since you sell the cupcakes for $5 each, the revenue function for this problem is given by:
R(q) = 5q
General equation for the cost function and the cost function in context:
The cost function is the sum of the variable costs (i.e., the product of the marginal cost and quantity) and the fixed costs:
Cost = (marginal cost * quantity) + fixed cost:
Thus, the general equation of the cost function is given by:
C(q) = mq + b, where
- C is the cost per q items produced,
- m is the change in cost per additional items produced,
- and b is the cost even when 0 items are produced (e.g., the cost to rent a booth).
Since it costs $1 to make each cupcake and $100 to rent the booth (you pay this cost even if you make 0 cupcakes), the cost function for this problem is given by:
C(q) = q + 100
Finding the value of q that makes R(q) and C(q) equal:
Now we can determine how many cupcakes must be sold for the expenses to be equal by setting R(q) equal to C(q) and solving for q:
R(q) = C(q)
(5q = q + 100) - q
(4q = 100) / 4
q = 25
Thus, you must make and sell 25 cupcakes to equal the expenses.