"Published in Newark, California, USA"
A farmer estimates that if he digs his potatoes now, he will have 120 bushels, which he can sell at $1.75 per bushel. If he expects his crop to increase 8 bushels per week, but the price to drop 5 cents per bushel per week, in how many weeks should he sell to realize the maximum amount for his crop?
Solution:
Let x be the number of bushels of potatoes per week
Let y be the price change of potatoes per bushels per week
Let t be the time/period of harvesting potatoes in weeks
Let C be the total cost of the potatoes
By analyzing the problem above, initially, there are 120 bushels of potatoes that the farmer can sell at $1.75 per bushel.
Total Cost of Potatoes = (Total Number of Potatoes in bushels)(Total Price of Potatoes per bushel)
C (initial) = (120)($1.75)
At the next statement, his crop will increase 8 bushels per week. The total number of potatoes in bushels can be written as
Total Number of Potatoes = 120 + xt
and there will be a change of price of potatoes per bushels per week. The total price of potatoes per bushel can be written as
Total Price of Potatoes per bushel = $1.75 + yt
Therefore, we can now write the working equation of Total Cost of Potatoes as follows:
Next we have to take the derivative of the above equation with respect to t, we have
You notice that we use the derivative of the product of two functions. Next, set dC/dt = 0 because we want to maximize the total cost of potatoes.
Therefore, he should sell his potatoes in 10 weeks.