Linda decides to open a kiosk in the mall selling baseball hats. It costs her $2280 to stock 100 hats and $3580 to stock 500 hats. She decides to charge $8 for every hat that she sells. 1) What is the marginal cost for each bottle? 2) What are Claire’s fixed costs to set up the kiosk? 3) Write Claire’s cost function. 4) Write Claire’s revenue function. 5) How many bottles of lotion does Claire’s have to sell in order to break even? 6) Write Claire’s profit function. 7) What is Claire’s profit if she sells 1,000 bottles?

The fixed cost would be $2280 because it is the cost that is required for setting up the kiosk. The cost $3580 is not relevant because it depends on the demand of the product. The least cost to set up kiosk is $2280.

3. Cost Function

Total Cost = Fixed Cost + Variable Cost

As we know that:

Variable Cost = Marginal cost per unit * Number of units = $3.25 * x = 3.25x

For Fixed cost $2280

By putting this value in the above equation, we have:

Total Cost = $2280 + 3.25x

C(x) = $2280 + 3.25x

And

For Fixed cost $3580

C(x) = $3580 + 3.25x

4. Revenue Function

Total Revenue = Selling Price per Unit * Total Units

Here

Selling price is $8 and total units are "x"

By putting values, we have:

Total Revenue = $8 * x

R(x) = 8x

5. Breakeven Point For $2280 and $3580

As we know that

Breakeven Point = Fixed Cost / Contribution Per unit

For Fixed Cost $2280:

Breakeven Point = $2280 / ($8 - $3.25)

= 480 Units

For Fixed Cost $2280:

Breakeven Point = $3580 / ($8 - $3.25)

= 754 Units

6. Profit Function

For Fixed Cost $2280:

Profit = Revenue Function - Cost Function

P(x) = 8x - ($2280 + 3.25x)

P(x) = 8x - $2280 - 3.25x

P(x) = 4.75x - $2280

For Fixed Cost $3580:

P(x) = 4.75x - $3580

7. Claire's Profit if she sells 1,000 bottles

Using the above profit function for fixed cost $2280, we have:

P(x) = 4.75x - $2280

Here x is 1,000 units, which means:

P(x) = 4.75 * 1,000 - $2280

P(x) = $4,750 - $2280 = $2,470

Using the above profit function for fixed cost $3,580, we have:

P(x) = 4.75x - $3,580

Here x is 1,000 units, which means:

P(x) = 4.75 * 1,000 - $3,580

P(x) = $4,750 - $3,580 = $1,170