Question

Bus Econ 1.2.35 * :Ξ Question Help Joanne sells silk-screened T-shirts at community festivals and craft fairs. Her marginal cost to produce one T-shirt is $5.50. Her total cost to produce 70 T-shirts is $445, and she sells them for $9 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) 60+ 5.5x. b. Joanne must produce and sell 110 T-shirts in order to break even.

Answer 1

First, we need to define marginal cost, which is how much it takes to produce one additional shirt (that is, one more shirt than is being produced now).

We define a function of C (cost) to be dependent on a base cost "a" (which it costs to make no shirts at all, presumably to pay for overhead in the factory and such), a marginal cost "m", and a number of shirts "s". In this case, it is linear.

(a)
C = ms + a
C = 5.50 s + a
We need to find a, the base cost, which we can do given a condition: "Her total cost to produce 70 t-shirts is $445":
445 = 5.50 * 70 + a
We solve for a, which turns out to be 70.
So, the full equation is C = 5.50 s + 70

(b)
This requires having a function for profit, simply the revenue (income) minus the production costs (given by the cost function C)

P = R - C
She sells shirts at 9 apiece so R = 9 s

So, P = 9s - 5.50 s - 70
Simplified, P = 3.50 s - 70

In order for her to break even (i.e., for profit to be zero), we set P=0 and solve for s, which ends up as 20 shirts. we say she needs to sell 20 shirts to break even.

(c)
We set P=600 and solve for s. We get 191.428, and round up to 192, since we can't have fractions of shirts.