Question

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 9 + N2 where k is a positive constant. What nitrogen level gives the best yield?

Answer 1

Answer:

• It is 3.

Explanation:

The model written correctly is:

• $Y=\frac{kN}{9+N^2}$

This is a mathematical question, instead of a chemistry question, and you should use calculus to find the nitrogen level that gives the best yield, since this is an optimization problem.

The best yield is the maximum yield, and the maximum, provided that it exists, is found using the first derivative and making it equal to zero: Y' = 0

To find Y' you must use the quotient rule.

$Y'=\frac{(kN)'(9+N^2)- (kN)(9+N^2)'}{(9+N^2)^2}\\ \\Y'=\frac{k(9+N^2)-kN(2N)}{(9+N^2)^2}\\ \\Y'=\frac{9k + kN^2 - 2kN^2}{(9 + N^2)^2}=\frac{9k-kN^2}{(9+N^2)^2}$

Now make Y' = 0

• The denominator is never equal to zero, because it is always positive and greater than 9.

• Make the numerator equal to zero:

         9k - kN² = 0

• Factor: k (9 - N²) = 0

• Since k is a positve constant, it is not equal to zero, and the other factor, 9 - N², must be equal to zero:

         9 - N² = 0 ⇒ (3 - N) (3 + N) = 0

         ⇒ 3 - N = 0 or 3 + N = 0 ⇒ N = 3 or N = -3.

Since N is nitrogen level, it cannot be negative and the only valid answer is N = 3.

You can prove that it is a maximum (instead of a minimum) finding the second derivative or testing some points around 3 (e.g. 2.5 and 3.5).