SOLUTION: Two numbers have a sum of 36. Write an equation for the product of the two numbers. Use roots to explain why the maximum product must occur when the two numbers are both 18.

Click here to see ALL problems on Quadratic Equations

Question 1179956: Two numbers have a sum of 36.
Write an equation for the product of the two numbers.
Use roots to explain why the maximum product must occur when the
two numbers are both 18.
Found 3 solutions by MathLover1, greenestamps, ikleyn:
Answer by MathLover1(20849)   (Show Source):
You can put this solution on YOUR website!
Let the number be and;
Given that the sum is
==>
We will write as function of :
==> .............(1)
Now we need to find the numbers such that their product is a maximum.
Let be the product:
==>
But
==>
==>
Now we need to find the maximum point of
Since the sign of is negative, then the function has a
find roots:
==>
==>
==> real solutions are or
disregard solution, so
==>
then go to
.............(1), substitute
==>
Then the numbers are and and the maximum product is:

Answer by greenestamps(13198)   (Show Source):
You can put this solution on YOUR website!

Given that the sum of the two numbers is 36, the solution from the other tutor uses the typical algebraic method, calling the numbers x and 36-x.

Here is another approach....

Let the two numbers be 18+x and 18-x; their sum is 36, as required.

The product of the two numbers is then (18+x)(18-x)=324-x^2.

In that form, the product is clearly a maximum when x=0, making the two numbers 18+0=18 and 18-0=18.

Answer by ikleyn(52778)   (Show Source):
You can put this solution on YOUR website!
.

All these statements rotate around one well known fact:

        if the perimeter of a rectangle is given, then
        its area is maximum when the rectangle is a square.

See the lessons
    - A rectangle with a given perimeter which has the maximal area is a square
    - A farmer planning to fence a rectangular garden to enclose the maximal area
in this site.