Question

A commercial apple grower estimates from past records that if 50 trees are planted per acre, then each tree will yield an average of 70 pounds of apples per season. If, for each additional tree planted per acre (up to 30), the average yield is reduced by 1 pound, how many trees should be planted per acre to obtain the maximum yield per acre? What is the maximum yield?

Answer 1

i) 60 trees

ii) 3600 apples

Explanation:

50 trees per acre

Total pounds produced per acre ( ideally ) = 70 * 50 = 3500 pounds

Assumption :

lets assume the additional tree = X

Therefore the number of trees planted per acre = ( 50 + x ) where x ≤ 30

calculate average yield per tree = ( 70 - x )

Total yield per acre = number of trees * average yield per tree

= ( 50 + x ) * ( 70 - x )

Y(x) = 3500 + 20x - x^2

i) Number of trees that should be planted per acre to obtain maximum yield per acre

y'(x) = 20 - 2x = 0 hence X ( number of additional trees ) = 20 / 2 = 10

Number of trees that should be planted per acre = 50 + 10 = 60 trees

ii) Calculate maximum yield

where ; X = 10

number of trees planted per acre = ( 50 + x ) = 50 + 10 = 60

hence maximum yield = ( 50 + x ) * ( 70 - x ) = ( 60 ) * ( 60 ) = 3600 apples