Finding the linear cost

Question

Joanne sells? silk-screened T-shirts at community festivals and craft fairs. Her marginal cost to produce one? T-shirt is $ 3.50. Her total cost to produce 50 ?T-shirts is $ 255, and she sells them for ?$9 each.
a.) Find the linear cost function for Joeannes t-shirt production.
b.)How many? T-shirts must she produce and sell in order to break? even?
c.How many? T-shirts must she produce and sell to make a profit of ?$500?

Answer 1

Hi Abigail.

 

Let's set this up.

 

a)  Our equation is of the form y=mx+b

y = total cost

x = # t-shirts

m = marginal cost 

b = fixed cost.

Entering the data we have, based on producing 50 t-shirts.

255 = 3.5(50) + b

We can solve for b

255 = 175+b

b = 80

 

So, our linear cost function is

y = 3.5x + 80

 

b)  Joanne sells her shirts of $9 each.

9x = the revenue from selling x shirts

Since we want to break even

9x = 3.5x + 80

 

By solving for x, we'll have the number of shirts needed to break even.

5.5x = 80

x = 14.55

Since you can't make and sell a partial shirt, Joanne will have to make and sell 15 shirts to break even.

 

c) Making a profit of $500

9x - (3.5x+80) = 500

5.5x - 80 = 500

5.5x = 580

x = 105.45

Again, we can't have a partial shirt, so Joanne will need to produce and sell 106 shirts in order to make a $500 profit.