Joanne sells silk-screened T-shirts at community festivals and craft fairs. Her

Question

Joanne sells silk-screened T-shirts at community festivals and craft fairs. Her marginal cost to produce one T-shirt is $4 50 Her total cost to produce 50 T-shirts is $305, 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? a. The linear cost function is C(x) = 4.5x + 80. b. Joanne must produce and sell 23 T-shirts in order to break even? c. Joanne must produce and sell 194 T-shirts to make a profit of $600.

Explanation / Answer

a. Since marginal cost is 4.5,

Integrating,

linear cost = 4.5x + c   

When x = 50

=> 305 = 4.5*50 + c

=> 305 = 225 + c

=> c = 80

So the linear cost function is 4x + 80

Selling price = $8

b. Let the number of t-shirts for break-even be y

So 8y = 4.5y + 80

=> 3.5y = 80

=> y = 80/3.5 = 22.86

So she should sell 23 t-shirts to break even.

c. Let z be the number of t-shirts sold to make a profit of $600.

Total selling price of t-shirts = 8z

So cost price = 8z - 600

=> 4.5z + 80 = 8z - 600

=> 3.5z = 680

=> z = 680/3.5 = 194.28

Thus she must produce and sell 195 t-shirts.

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