What is the Maximum Product of Two Numbers that Sum to 120?

[whitehorsey]

1. Find two numbers x and y whose sum is 120 and whose product is a maximum.



2. none



3. x + y = 120
xy = maximum
x = 120 - y
I wrote these down but I don't know what to do next. Can you please show me? :) Thank You!

 

[danago]

Well you have the function P = xy which you want to maximize, however it contains two variables. Are they independent though? How can you reduce this function down to a single variable function?

 

[HallsofIvy]

1. Find two numbers x and y whose sum is 120 and whose product is a maximum.



2. none



3. x + y = 120
xy = maximum
x = 120 - y
I wrote these down but I don't know what to do next. Can you please show me? :) Thank You!

Replace the x in xy with 120- y so you have only one variable. Now, do you know how to "apply the derivative" to find where a function has a maximum or minimum?

 

[Mark44]

As an alternate approach, the graph of your area function, as a function of either x or y alone, is a parabola. You can find the vertex of the parabola, at which the maximum area is attained, without using calculus.

 

[HallsofIvy]

Ah! Excellent point.