54.7k views
1 vote
Your baking club is selling cupcakes. It costs 100 to rent the booth. If the cost to make each cupcake is $1 and each cupcake is sold for $5 how many cupcakes need to be sold to equal the expenses

2 Answers

1 vote

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.

User Casey Wilkins
by
8.2k points
2 votes

Answer:

You need to sell 25 cupcakes

Explanation:

let y be the total cost of making cupcakes + the rent

let x be the number of cupcakes

y = x + 100 Equation1

You sell each cupcake for $5

y = 5x Equation2

substitute eq.1 to eq2

x + 100 = 5x

4x = 100

x = 25

∴ You need to sell 25 cupcakes to equal the expenses

let's check

25($1) + $100 = 25($5)

$25 + $100 = $125

$125 = $125

User Nese
by
7.9k points
Welcome to QAmmunity.org, where you can ask questions and receive answers from other members of our community.

9.4m questions

12.2m answers

Categories