A model used for the yield y of an agricultural crop as a function of the nitrogen level n in the soil (measured in appropriate units) is y = kn 1 + n2 where k is a positive constant. what nitrogen level gives the best yield?

Question

Physics

Mikaela Simmons

15 December, 10:07

A model used for the yield y of an agricultural crop as a function of the nitrogen level n in the soil (measured in appropriate units) is y = kn 1 + n2 where k is a positive constant. what nitrogen level gives the best yield?

+1

Answers (1)

Know the Answer?

Oscar Everett · Answer 1

The maxima of an equation can be obtained by taking the 1st derivative of the equation then equate it to 0. The value of N that result in best yield is when dy/dn = 0.

Taking the 1st derivative of the equation y = (kn) / (9+n^2):

By using the quotient rule the form of the equation is:

y = g (n) / h (n)

where:

g (n) = kn - - - > g' (n) = k

h (n) = 9 + n^2 - - - > h' (n) = 2n

dy/dn is defined as:

dy/dn = [h (n) * g' (n) - h' (n) * g (n) ] / h (n) ^2

dy/dn = [ (9 + n^2) (k) - (kn) (2n) ] / (9 + n^2) ^2

dy/dn = (9k + kn^2 - 2kn^2) / (9 + n^2) ^2

dy/dn = (9k - kn^2) / (9 + n^2) ^2

dy/dn = k (9 - n^2) / (9 + n^2) ^2

Equate dy/dn = 0, then solve for n

k (9 - n^2) / (9 + n^2) ^2 = 0

k (9 - n^2) = 0

9 - n^2 = 0

n^2 = 9

n = sqrt (9)

n = 3

Answer: The nitrogen level that gives the best yield of agricultural crops is 3 units.