1. Calculus economics problem

A craftsperson has the capacity to produce up to 16 pieces of ceramic pottery per week. Experience has shown that x pieces per week can be sold at a price of (140-2x) each and the cost of producing x pieces is (600+20x+x^2). How many pieces of pottery should be made each week to give the greatest profit?

no clue what to do

2. Re: Calculus economics problem

$profit = sales - cost = x(140-2x) - (600+20x+x^2) = 140x -2x^2 -600 -20x -x^2 = -(3x^2 - 120x + 600) = -3(x^2 -40x+200)$

we want to maximize the profit so we set it's first derivative to zero and solve.

$\dfrac{d}{dx} profit = -6x + 120$

$-6x+120 = 0 \Rightarrow x = 20$

well the craftsperson can't produce 20 a day

but we see that the derivative is positive for $x < 20$ so the maximum profit will be attained with the maximum number of pieces they can produce a day which is 16.