Answer
a) $ C(x)=3.5x+90$
b) $17$
c) $108$
Work Step by Step
a) Marginal cost = $3.5$ dollar/T-shirt
Selling Price = $9$ dollar/T-shirt
let the cost function $C(x)$ be $C(x)=ax+b$ where $a=3.5$
It is given that the total cost for producing $60$ T-shirts is $300$. i.e. when $x=60, C(x)=300$
$300=(3.5\times60)+b\implies b=90$
Therefore, $C(x)=3.5x + 90$
b) She sells T-shirts for dollar $9$ each.
$S(x)=9x$
Break even means she doesn't profit anything at all, i.e. Profit = $S(x)-C(x)=0$.
$S(x)=C(x)$
$9x=3.5x+90$
$x=16.36\approx 17$
She must produce and sell $17$ T-shirts.
c) Profit =$500$ dollars
$S(x)-C(x)=500$
$9x-3.5x-90=500$
$5.5x=590$
$x=107.27\approx 108$
She must produce and sell $108$ T-shirts to make a profit of $500$ dollars.
