Max product of two numbers whose sum is 30?

[lcliff]

I need help solving a problem. The original question was: What is the maximum product of two numbers whose sum is 30? To solve the problem, I used excel and listed numbers 0 to 30 and then added them to numbers going down from 30 to 0, then I multiplied those coming up with 225.

The instructor says that we (the class has all done it pretty much the same way) need to derive the answer mathematically, not by process of elimination.

Any advice??

 

[stapel]

Pick variables for the two numbers. Create an equation representing their given sum value. Solve this equation for one of the variables (it doesn't matter which one you pick).

Using the other variable and the expression for the solved variable, create the product expression. This will give you a negative quadratic.

The maximum value will be found at the vertex.

Eliz.

 

[wjm11]

What is the maximum product of two numbers whose sum is 30?

Stapel beat me to the punch , but here's what she's telling you:

Consider:
x + y = 30
So, y = 30-x

x*y = c (some number, the one you wish to maximize)
(30-x)*x = c (by substitution)
-x^2 + 30x = c

This is the equation of a parabola that opens “downward” if you plot c on the vertical axis and x on the horizontal axis. If you plot this curve, you will find that c is a maximum at the vertex of the parabola. The x value of the vertex point is:

x = -b/2a, or in this case, where a = -1 and b = 30,
x = -(30/[2(-1)]) = 15 (which you already knew, as 15^2 = 225)

 

[Denis]

In case this helps:
you can look at 30 as being the width + length of a rectangle that produces maximum area.

 

[lcliff]

Thank you three so much! This really helps to put it in perspective for me!