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Joanne sells​ silk-screened T-shirts at community festivals and craft fairs. Her marginal cost to produce one​ T-shirt is $ 2.50 . Her total cost to produce 60 ​T-shirts is $ 240 comma 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 ​$600​?

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sqdancefan

Answer:

  a) c = 90 +2.50t

  b) 17

  c) 126

Step-by-step explanation:

a) We're told that the cost for 60 T-shirts is $240, but we know that the marginal cost for those 60 shirts is ...

  60($2.50) = $150

Joanne seems to have fixed costs of $240 -150 = $90. So, her linear cost function seems to be ...

  c = 90 +2.50t . . . . . . . where c is the cost to produce t shirts

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b) Joanne's revenue function is ...

  r = 8t

so her profit function is ...

  p = r -c = 8t -(90 +2.50t)

  p = 5.50t -90

This will be zero for ...

  0 = 5.50t -90

  0 = t -16.364

Joanne must produce and sell 17 T-shirts to cover her production cost.

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c) Profit will be 600 when ...

  600 = 5.50t -90

  690 = 5.50t

  690/5.50 = t = 125.455

Joanne must produce and sell 126 T-shirts to make a profit of $600.

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Comment on results

Note that fractional T-shirt sales are involved in making the numbers come out exactly. We have elected to round up, so that slightly more profit is made than the exact amounts of $0 or $600. Rounding to the nearest integer would give values one shirt less than what we have reported.

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