Question

A bicycle store costs ​$3750 per month to operate. The store pays an average of ​$45 per bike. The average selling price of each bicycle is ​$95. How many bicycles must the store sell each month to break​ even?

Answer 1

The store must sell at least 75 bikes each month to break even.

Explanation:

The break-even point is when revenue equals cost.

The revenue function is R(q) = pq, where

• R(q) is the revenue,
• p is the price of an item,
• and q is the quantity of an item.

Since we're told that the average selling price of each bicycle is $95, our revenue function is R(q) = 95q

The cost function is C(q) = mq + b, where

• C(q) is the cost,
• m is the marginal cost (change in cost per one additional item made), and
• b is the fixed costs (costs paid without making anything).

Since we're told that the store pays an average of $45 per bike, 45 is our marginal cost. Since the store costs $3750 per month to operate, the fixed costs is 3750.

Thus, our cost function is C(q) = 45q + 3750.

Now, we set our two functions equal to each to solve for q, the number of bikes the store must sell each month to break even:

C(q) = R(q)

45q + 3750 = 95q

3750 = 50q

75 = q

Thus, the store must sell at least 75 bikes each month to break even.

Answer 2

To break even, the store's revenue must be equal to its expenses. Let's set up an equation to solve for the number of bicycles the store must sell each month to break even.

Let's use x to represent the number of bicycles the store must sell each month to break even.

Revenue = Selling price per bike * Number of bikes sold = 95x

Expenses = Fixed cost + Variable cost per bike * Number of bikes sold = 3750 + 45x

Since the store must break even, we can set the revenue equal to the expenses:

95x = 3750 + 45x

Simplifying the equation:

50x = 3750

x = 75

Therefore, the store must sell 75 bicycles each month to break even.