What is the maximum product of two numbers when their sum is 100?

[Geekchick]

Homework Statement

Find two numbers whose product is a maximum if the sum of the first number and twice the second is 100

Homework Equations

The Attempt at a Solution

Alright so I think I might have the right answer but something just doesn't seem right.

so first I named my two variables x,y then I set up the problem in terms of x and y

x+2y=100

then I solved for y to get

y=50-x/2

then I multiply the new y value and x to get the maximum

x(50-x/2)=0 solve to get

x=25/2

so then I plug my x value back into get my y value and end up with

(25/2,175/4) as my maximum right?

 

[Geekchick]

oh wait so I found the answer in my textbook (the original question was on a lab but my teachers almost all the lab questions from the book) and the answer is 50 and 25 which makes sense but i still can't find the flaw in my logic : (

 

[statdad]

Why do you set

$$x\left(50 - \frac x 2 \right)$$

equal to zero and solve for $$x$$? You don't want this product to be zero, you want it
to be the maximum value possible. That is where your error lies.

 

[Geekchick]

oh yeah, you're right.

 

[Mark44]

Homework Statement

Find two numbers whose product is a maximum if the sum of the first number and twice the second is 100

You determined that the two numbers are x and 50 - x/2.

Let's get rid of y as you first defined it (i.e., as the other number, which you've already figured out) and let's now use it to represent the product of the two numbers.

So y = x(50 - x/2)

If you think of the graph that the equation above represents, what you found is where that graph crosses the x-axis. You found one of these points and missed the other one.

Is there a high point for this graph? That's really what you're looking for, not where the graph crosses the x-axis.