Best Way to Form an Equation On This Optimisation Question?

In summary: This is quadratic in form. The maximum value will be at the vertex. x = -b/2a = -800/-0.2 = 4000. 4000 additional trees will give the maximum total population of oranges.In summary, an orchardist can maximize the total population of oranges in their grove by planting 4000 additional trees, which will result in 50+4000=4050 trees and a total population of oranges of (800-0.10(4000))(4050) = 1,996,000 oranges.
  • #1
Procrastinate
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An orchardist has one grove of 50 orange trees. Each tree produces 800 oranges. The orchardist knows from experience that each addition tree planted in the grove will reduce the output of each tree by .10 oranges. How many tree should the orchardist plant in the grove in order to maximize the total population?



I am just wondering if there are any good tips on finding an equation for this question. I want to improve my problem solving skills but I can't understand how the answer was found (800-(x-50)x) or something similar (I don't have my solutions on me.)

I don't need to know how to solve it.

Thanks.
 
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  • #2
Procrastinate said:
An orchardist has one grove of 50 orange trees. Each tree produces 800 oranges. The orchardist knows from experience that each addition tree planted in the grove will reduce the output of each tree by .10 oranges.
So the number of oranges produced by each tree, if there are "x" more trees planted, is 800- .10x.

How many tree should the orchardist plant in the grove in order to maximize the total population?



I am just wondering if there are any good tips on finding an equation for this question. I want to improve my problem solving skills but I can't understand how the answer was found (800-(x-50)x) or something similar (I don't have my solutions on me.)

I don't need to know how to solve it.

Thanks.
If x more trees are planted, there will be 50+ x trees. The total number of oranges is, of course, "oranges per tree" times "number of trees"- (800- .10x)(50+ x).
 

Related to Best Way to Form an Equation On This Optimisation Question?

1. What is an equation?

An equation is a mathematical statement that shows the relationship between two or more variables. It is written using symbols, numbers, and mathematical operations such as addition, subtraction, multiplication, and division.

2. Why is it important to form an equation on an optimization question?

Forming an equation on an optimization question allows you to represent the problem mathematically and find the optimal solution. It helps to identify the variables involved and their relationships, making it easier to solve the problem.

3. How do I identify the variables in an optimization question?

To identify the variables, you need to carefully read the question and determine what quantities are changing and what quantities are constant. The changing quantities are usually the variables, while the constant quantities are the known values or constraints.

4. What are the steps for forming an equation on an optimization question?

The steps for forming an equation on an optimization question are:

  1. Read the question and identify the variables and constraints.
  2. Write an expression for the objective function, which represents the quantity you want to optimize.
  3. Write an equation that relates the variables and constraints to the objective function.
  4. Differentiate the equation and set the derivative equal to zero to find the critical points.
  5. Check the critical points and determine which one gives the optimal solution.

5. Are there any tips for forming an equation on an optimization question?

Yes, some tips for forming an equation on an optimization question are:

  • Read the question carefully and understand what is being asked.
  • Identify the variables and constraints.
  • Use clear and concise notation.
  • Write the equation in terms of a single variable, if possible.
  • Double-check your work to ensure that the equation is correct.

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