On a citrus farm the average annual yield of oranges is 160

On a citrus farm, the average annual yield of oranges is 160 pounds per tree when the number of trees is 20 per acre. For each additional tree over 20, the annual yield per tree decreases by 2 pounds per tree. How many orange trees should be planted per acre to maximize the number of pounds of oranges per acre? Number of orange trees

Solution

let the additional number of trees per acre=x

total number of trees per acre=20+x

yield per tree =160-2x

total yield per acre ,Y= (total number of trees per acre)*(yield per tree)

Y=(20+x)(160-2x)

Y=-2(20+x)(x-80)

Y=-2(x²-60x-1600)

Y=-2(x²-60x +(60_/2)²-(60_/2)²-1600)

Y=-2(x²-60x +30²-900-1600)

Y=-2(x²-60x +30²)+1800+3200

Y=-2(x-30)²+5000

yield is maximum when x=30

total number of trees per acre to be planted for maximum yield =20+30

total number of trees per acre to be planted for maximum yield =50