A farmer can get 2 dollars per sack of potatoes on the first of July and after t

Question

A farmer can get 2 dollars per sack of potatoes on the first of July and after that the price drops by 2 cents per sack per day. On July first, the farmer has 80 sacks of potatoes in the ground. She estimates that the crop is increasing by 1 sack per day. When should she harvest to get the most money? (Express your answer as the number of days after the first of July.)

A rectangular field will have one side made of a brick wall and the other three sides made of wooden fence. Brick wall costs 10 dollars per meter and wooden fence costs 15 dollars for 3 meters. The area of the field is to be 1350 m2. What length should the brick wall be to give the lowest total cost of wall plus fence?

Explanation / Answer

Let x= the number of days after july1

If x= 0, then her income is 80*2= 160 dollars
Then 80+x = number of sacks
2-.01x = price per sack

Income= (80+x)(2 - 0.01x)= 160 - 0.80x +2x - 0.01x^2

= -0.01x^2 + 1.2x +160

This is a parabola that opens down, so The maximum occurs at the vertex.

Without calculus, x= -b/(2a)= -1.2/(2*-.01)= 60

Using derivatives, y' = -0.02x+ 1.2= 0
X= -1.2/(-0.02) = 60

So she should harvest 60 days after July 1.

B) The greatest rectangle area for least perimeter is a square.
Area of 1350m^2, square side = SQR(1350) = 36.74m
Wood fence (15/5 = $3/m)
= 3(36.74)(3) = $330.66
Brick wall = 36.74(10) = $367.4

The brick wall should be 36.74 meters long.

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