A farmer has a herd of 100 cattle with an average weight of 500 lbs/cow. It costs 50 cents a day to keep each cattle. They are gaining weight at a rate of 6 lbs per day. The market price is now $1 per lb for beef and is falling by 1 cent each day. How long should the farmer wait to sell his cattle in order to earn the most money? How much has he gained by waiting rather than selling right now?

Answer 1

weight: w(t) = 500+6t
cost: c(t) = 0.50t
price: p(t) = 1.00 - 0.01t
sales profit per cow: s(t) = w(t)*p(t)-c(t)
s(t) = (500+6t)(1-t/100)- t/2

You can ignore the number of head; that's just a constant multiplier.

s(t) is just a parabola, so find the vertex for maximum profit.