Phil makes and sells rugs at his roadside stand His monthly

Phil makes and sells rugs at his roadside stand. His monthly fixed cost for owning the stand is $690. If he makes and sells 26 rugs, his total costs are $846 and he brings in $546 in revenue. Find Phil\'s monthly cost, revenue, and profit functions (assuming they are linear). Let x be the number of rugs made and sold each month.

Solution

If his fixed cost is $690 and his total costs for 26 rugs is $846, then his marginal cost per rug is
($846 - $690) / 26 = $156/26 = $6

Thus his costs (in dollars) as a function of x, the number of rugs produced, are
C(x) = 6x + 690

If 26 rugs bring in $546 in revenue, then the revenue per rug is
$546 / 26 = $21
and the revenue function in dollars is
R(x) = 21x

The profit is revenue minus cost, so the monthly profit in dollars is
P(x) = R(x) - C(x)
= 21x - (6x + 690)
= 15x - 690

Of course, his break-even point is the number of rugs per month required to achieve $0 profit, which is 46.

That is 15x - 690 = 0

15x = 690

x = 690 / 15 = 46