SOLUTION: Find two positive numbers whose sum is 8 and the product is minimum.

Question 1208681: Find two positive numbers whose sum is 8 and the product is minimum.
Found 2 solutions by greenestamps, math_tutor2020:
Answer by greenestamps(13200) (Show Source): You can put this solution on YOUR website!

There is no solution to the problem as posed.

Given the sum of two positive numbers, the product is a MAXIMUM when the two numbers are the same; there is no minimum value of the product.
x 8-x x(8-x) ----------------------- 1 7 7 .1 7.9 0.79 .01 7.99 0.0799 .001 7.999 0.007999 etc...

Answer by math_tutor2020(3816) (Show Source): You can put this solution on YOUR website!

As tutor greenestamps points out, there is no minimum since the results get closer to 0.
I'll assume your teacher meant to say "maximum" instead of "minimum".

Let x be one of the numbers.
8-x must be the other number if we want them both to add to 8.
x+(8-x) = 8.

Their product is x(8-x) = -x^2+8x

This is a parabola that opens downward.

You can use graphing tools like GeoGebra or Desmos (among many other tools) to confirm this is the correct graph.
If you are familiar with a TI83, then it's probably best to use that instead.

The key thing to note on the graph is the vertex (4,16)
Here's how we find the coordinates of that location.

Consider the parabola y = ax^2+bx+c.
Its vertex is located at (h,k) where h = -b/(2a)
In the case of y = -x^2+8x we have a = -1, b = 8, c = 0 (side note: since a < 0, the parabola opens downward to produce a highest point).

Let's plug in those values:
h = -b/(2a)
h = -8/(2*(-1))
h = 4
This x coordinate of the vertex.
Another way to find this value is to apply the midpoint formula to the roots x = 0 and x = 8. This works due to the parabola's symmetry.

That x value is then plugged into the equation to find its paired y coordinate.
y = -x^2+8x
y = -4^2+8*4
y = -16+32
y = 16
Or you could say
y = x*(8-x)
y = 4*(8-4)
y = 4*4
y = 16
This would tell us that the largest possible product of x and 8-x is 16, and it happens when both values are 4.

------------------------------
A real world application:
A famer has 16 meters of fencing and wishes to form the largest rectangle in terms of area.
The farmer should make a 4 by 4 square to max out the area.
perimeter = 16 meters ---> side length = 16/4 = 4 meters ----> area = 4^2 = 16 square meters

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