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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$$ Y = \dfrac{kN}{1 + N^2} $$where $ k $ is a positive constant. What nitrogen level gives the best yield?

1

01:36

Wen Z.

00:47

Amrita B.

Calculus 1 / AB

Calculus 2 / BC

Chapter 4

Applications of Differentiation

Section 7

Optimization Problems

Derivatives

Differentiation

Volume

Missouri State University

Campbell University

Oregon State University

University of Nottingham

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were given a model used for the yield. Why of an agricultural cop crop as a function of the nitrogen level end in the soil. This model is why equals K Times n over one plus n squared for K is some positive constant were asked what nitrogen level will give us the best yield? Do this. We want to find the derivative. Why prime? So why Prime is a function of n This is the bottom one, plus n squared times the derivative of the top. Okay, right at the top K n times the drop of the bottom which is to end our over the bottom one plus and squared, I swear. Jesus, I like work. This is equal to factor out a k we get k times one plus n squared minus 21 squared is one minus and squared over one plus and squared. Squared? Yeah. No. Yeah, but now we find the critical points by setting this equal to zero. So we get one minus and squared, equals zero. Or in other words, and is equal to plus or minus one. What you get now, we can't have negative yield. This doesn't really make any sense. So we're really just looking at an equals positive one, right? Well, it's nice. Wrong hole. Don't see ends at the Children's Hospital. All the ends had to go for the war effort in North Africa. Now we noticed that Why then approaches zero as n approaches infinity. Yes, yes. And therefore yeah, it follows that Why has in absolute impeached land parcel maximum on 02 infinity? Oh, it is like Yeah, So we're going to have to compare the values of why at our critical point and equals one and at the boundary zero So why of zero? This is simply zero, whereas y of n I'm sorry. Why of one? Eventually groups is equal to K over to which is, of course, greater than zero K is positive, to be sure. So it follows that about what years? Why achieves It's maximum value at n equals k over to this hour. Sorry n equals one screeching

Harvey Mudd College

Baylor University

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