Question 242803: Joanne sells silk-screened T-shirts at community festivals and craft fairs. Her marginal cost to produce one t-shirt is $450. Her total cost to produce 80 t-shirts is $420, and she sells them for $8 each.
a) find the linear cost function for joanne's 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 $900?
I need to know how to do it step by step, if anyone can help!! Thank you so much
Found 2 solutions by oberobic, stanbon:
Answer by oberobic(2304)   (Show Source):
You can put this solution on YOUR website! First, define your terms.
MC = marginal cost to make 1 t=shirt, which has to be $4.50 (not $450).
The cost to print 80 shirts is 420.
80*4.5 = 360, so there is an additional cost of $60.
Since this is not a marginal cost, it must be a fixed cost.
Recall that Total Cost = Fixed Cost + (Marginal Cost * Number of items produced)
TC = FC + MC(X), where X is the number of items
TC = 60 + 4.5X (the answer to part a, the cost function)
Now we should note that $8 is the Price that she charges buyers.
P = 8
Profit = P*X -TC = 8X - 60 - 4.5X
Break even is the point where Profit = 0.
0 = 8X - 60 - 4.5X
Add 60 to both sided
60 = 8X - 4.5X = 3.5X
Dividing by 3.5
60/3.5 = X = 17 and a fraction, so she has to sell 18 shirts.
18 * 8 = 144 total revenue
60 + 4.5*18 = 141 total cost
So that's very close to break even. Close enough.
Continuing with the question of $900 profit...
900 = 8X - 60 - 4.5X (by substituting back into the equation)
Adding 60 to both sides:
960 = 3.5X
Dividing by 3.5.
274 (and a fraction) = X
Therefore, we propose that when she sells 274 shirts, she earns $900 profit.
Check your work!
Profit = Total Revenue - Total Cost
Profit = $8 per shirt * 274 shirts - $60 fixed cost - $4.50 * 274 shirts
Profit = 2192 - 60 - 1233
Profit = 2192 - 1293 = 899
That checks as close as we can get it.
Why did I ignore the fractions?
You cannot sell a fraction of a shirt.
Answer by stanbon(75887) (Show Source):
You can put this solution on YOUR website! Joanne sells silk-screened T-shirts at community festivals and craft fairs.
Her marginal cost to produce one t-shirt is $450.
Her total cost to produce 80 t-shirts is $420, and she sells them for $8 each.
a) find the linear cost function for joanne's t-shirt production
You have 2 points: (1,450), (80,420)
--
slope = (420-450)/(80-1) = -30/79
intercept: 450 = (-30/79)*1 + b
b = 450.38
C(x) = (-30/79)x+450.38
--------------------------------------
b) How many t-shirts must she produce and sell in order to break even?
income = cost
8x = (-30/79)x+450.38
(8+(30/79))x = 450.38
x = 53.75
Round up to 54 t-shirts
---------------------------------------
c) How many t-shirts must she produce and sell to make a profit of $900?
Solve 8x-[(-30/79)+459.38] = 900 for "x".
===============================================
Cheers,
Stan H.
|
|
|
