Joanne sells silk screened T shirts at community festivals

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 40 - T - shirts is $150, and she sells them for $7 each. Find the linear cost function for Joanne\'s T - shirt production. How many T - shirts must be produce and sell in order to break even? How many T - shirts must she produce and sell to make a profit of $700?..The linear cost function is C(x) = (Type your answer in slope - intercept form.) Jonne\'s must produce and sell T - shirts in order to break even. Jonne must produce and sell T - shirts to make a profit of $700.

Solution

If the marginal cost is MC(x) = 2.5, where x = the number of shirts, then the cost function is the integral of the marginal cost: C(x) = 2.5x + c To find the value of c, we know that C(40) = 2.5(40) + c = 150 which implies 100 + c = 150 => c = 50 a) So C(x) = 2.5x + 50 b) Let P(x) = 7x be the profit function. To break even, 2.5x + 50 = 7x implies 2.5x - 7x + 50 = 0 => -4.5x + 50 = 0 => 4.5x = 50 => x = 11 Hence Joanne must produce and sell 11 shirts to break even. c) To make a profit of 700, P(x) - C(x) = 700. Thus 7x - 2.5x - 50 = 700 => 4.5x = 750 => x = 167 Therefore Joanne must produce 167 shirts to make a profit of 700.