Maximizing the Sum of Two Numbers: An Algebraic Solution

In summary, to find two numbers whose sum is 20 and whose product is a maximum, you can represent the numbers as x and 20-x. The product of these numbers can be expressed as a parabola, with the maximum value at the vertex. To find the vertex, you can complete the square and use the axis of symmetry. The numbers that will satisfy this condition are 10 and 10.
  • #1
ben328i
23
0

Homework Statement


find two numbers whose sum is 20 and whose product is a maximum.


Homework Equations


the first number is X
the second number is 20-x



3. The solution
X(20-X)=0
-X^2+ 20x=0
x=-b/2a = -20/2(-1) = 10
20 - x =20 -10 = 10

the numbers are 10 and 10


i just don't get why / how you know to put x and 20 - x and why you would use the axis of symmetry to find the numbers

and sry mods i posted originally in the wrong thread.
 
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  • #2
From the problem statement you have 2 numbers which sum to 20, that is x and 20-x.

It should be obvious that ( x )+ (20 -x) = 20 so you have represented the 2 numbers in general. Now you need to find when the product x(20-x) is a maximum.

Now if you were in a calculus class you would take the derivitive and set it to zero. Since you are not doing this I will have to assume that you are not in calculus. You have the problem of finding the maximum of the parabola, using properties of a parabola. The maximum will lie on the axis of symetry of the parabola, so all you need do is find the point on the parabola which lies on the symetry axis.
 
  • #3
thanks
not in calc but next year trig then pre and then calc
 
  • #4
X(20-X)=0 is not true. You have the function 20X- X2 which is a parabola with maximum value at its vertex. You can find the (X,Y) coordinates of the vertex by completing the square.
 

1. What is the purpose of maximizing the sum of two numbers?

The purpose of maximizing the sum of two numbers is to find the largest possible value that can be obtained by adding two numbers together. This can be useful in a variety of situations, such as maximizing profits or finding the maximum possible score in a game.

2. How is this problem solved algebraically?

This problem can be solved algebraically by using variables to represent the two numbers, setting up an equation to represent the sum of the two numbers, and then using algebraic techniques such as factoring or substitution to find the maximum value of the sum.

3. What are the key concepts needed to solve this problem?

To solve this problem, it is important to understand basic algebraic concepts such as equations, variables, and factoring. It is also helpful to have a strong understanding of mathematical operations such as addition, subtraction, multiplication, and division.

4. Can this problem be solved using other methods besides algebra?

Yes, this problem can also be solved using geometric methods, such as finding the maximum value of the sum of two numbers using a graph or geometric shape. Additionally, it can be solved using numerical methods, such as trial and error or using a computer program to test different values.

5. Are there any real-world applications of maximizing the sum of two numbers?

Yes, there are many real-world applications of this problem. For example, a business owner may use this concept to maximize profits by determining the optimal prices for their products. In sports, coaches may use this to determine the best lineup to maximize their team's score. It can also be applied in fields such as engineering and finance to find the maximum possible value for a given scenario.

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