Maximizing the Sum of Two Numbers: An Algebraic Solution

[ben328i]

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.

 

[Integral]

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 derivative 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.

 

[ben328i]

thanks
not in calc but next year trig then pre and then calc

 

[HallsofIvy]

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.