Question

an orchard owner has maintained records that shows that, if 25 apple trees are planted in one acre then each tree yields an average of 500 apples. the yield decreases by 10 apples per tree for each additional tree that is planted. how many trees should be planted for maximum total yield?

Answer 1

x=additional trees planted
total yield = numtrees*yield per tree
numtrees = x+25
yield per tree = 500-10x
total yield = (x+25)(500-10x)
to maximize yield, differentiate and set equal to 0
y' = (x+25)*-10 + (500-10x) = 0
10(x+25) = 500-10x
20x = 250
x = 12.5
to maximize yield 25+12.5 trees should be planted