Question

Tom wants to open a store that sells hats. He thinks he can sell them for $20 each. He estimates his monthly expenses to be $3000, and it will cost him $8 for each hat. Tom wants to analyze his estimated revenues and expenses to see how many hats he would need to sell each month in order to break even. He uses x to represent the number of hats and y to represent the total dollars. Write a function that represents Tom's expenses each month.Write a function that represents Tom's revenue if he sells the hats for $20 each.

Answer 1

Linear Models

It's a common practice to model real situations with linear equations.

The problem of cost, revenue, and profit for a business is a clear example of such a situation.

Let's analyze the store where Tom wants to sell hats. He calls x to the number of hats he expects to sell.

The costs involved in keeping his store up and running are (on a monthly basis):

* $3000 fixed. No matter if he sells or not

* $8 for each hat he offers

If he produces x hats, then the total costs are:

C(x) = 3000 + 8x

On the other hand, the income (revenue) of each hat sold is $20, thus for x hats, the revenue function is:

R(x) = 20x

The 'break-even' point is the value of x that makes the costs equal to the revenue. It's a no-loss no-profit situation.

Equating both functions:

3000 + 8x = 20x

We need to solve for x. Subtracting 8x:

3000 = 12x

Dividing by 12:

x = 3000 / 12

x = 250

Tom needs to sell 250 hats to break even.

If 251 or more hats are sold, Tom would start to have a positive profit