How do you solve this optimization 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)/(1+N^2)#

where k is a positive constant. What nitrogen level gives the best yield?
#N=?#

Answer 1

#N=1#

Take the first derivative with respect to #N:#
#y'=((1+N^2)k-kN(2N))/(1+N^2)^2#
#y'=(k+kN^2-2kN^2)/(1+N^2)^2#
#y'=(k-kN^2)/(1+N^2)^2#
Equate to #0# and solve for #N#:
#(k-kN^2)/(1+N^2)^2=0#
#k(1-N^2)=0#
#1-N^2=0#
#N^2=1#
#N=+-1->N=1# is the only possible answer as we cannot have a negative nitrogen level.
The "best yield" would entail #y# being at its maximum. To ensure that #N=1# gives a maximum for #y#, evaluate #y'# in the following intervals:
#[0, 1), (1, oo)# to determine whether #y'# is positive (#y# is increasing) or #y'# is negative (#y# is decreasing) in each interval.
If #N=1# is a maximum, then #y'# will be positive before we reach #N=1# and negative afterwards:
#[0,1):#
#y'(0)=(k-k(0))/1^2=k>0# So, #y# is increasing on #[0, N)#
#(1, oo):#
#y'(2)=((k-4k)/(1+4)^2)=-(3k)/25<0# So, #y# is decreasing on #(1, oo)# and the maximum possible crop yield happens with #N=1#.
Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer 2

To solve an optimization question, follow these general steps:

  1. Understand the Problem: Carefully read and understand the problem statement. Identify what needs to be optimized and any constraints given in the problem.

  2. Define Variables: Assign variables to represent the quantities involved in the problem. Clearly define what each variable represents and any constraints they must satisfy.

  3. Formulate the Objective Function: Write an expression that represents the quantity to be optimized. This could be maximizing or minimizing a certain value.

  4. Formulate Constraints: If there are any restrictions or limitations on the variables, formulate them as constraints. These constraints should be included in the problem formulation.

  5. Optimization Technique: Depending on the nature of the problem (linear, nonlinear, constrained, unconstrained), choose an appropriate optimization technique. Common techniques include calculus methods like differentiation, Lagrange multipliers, or linear programming methods.

  6. Solve for Optimal Solution: Use the chosen optimization technique to find the optimal solution. This may involve finding critical points, solving equations, or using optimization algorithms.

  7. Check Constraints: After obtaining a potential solution, ensure that it satisfies all given constraints. If not, revise the solution accordingly.

  8. Interpret Results: Once you have the optimal solution, interpret what it means in the context of the problem. Explain what the solution represents and its practical implications.

  9. Verify Solution: Finally, double-check your calculations and reasoning to ensure that your solution is correct and makes sense in the context of the problem.

By following these steps, you can effectively solve optimization problems across various disciplines and scenarios.

Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

Not the question you need?

Drag image here or click to upload

Or press Ctrl + V to paste
Answer Background
HIX Tutor
Solve ANY homework problem with a smart AI
  • 98% accuracy study help
  • Covers math, physics, chemistry, biology, and more
  • Step-by-step, in-depth guides
  • Readily available 24/7