SOLUTION: Why are these two similar problems set up differently? and why are they set up in that way at all? Find two numbers whose difference is 60 and whose product is a minimum? Why do

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Question 999941: Why are these two similar problems set up differently? and why are they set up in that way at all?
Find two numbers whose difference is 60 and whose product is a minimum? Why does this use f(x) = x(x-60)
and
Find two numbers whose sum is 60 and whose product is a maximum? Why does this one use f(x) = x(60-x)
I fundamentally don't understand what is happening and how that function was created at all.
And will there ever be a case where it's a difference and the product is a maximum. Or where it's a sum and the product is a minimum. I don't understand what this question is asking.
So confused...
Please help!
Found 2 solutions by Fombitz, MathTherapy:
Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!
Let the two numbers be x and y.
Assume x is larger than y.
The sum would be x%2By=60
So then y=60-x
So the product of the two numbers is,
P%5B1%5D=x%2Ay=x%2860-x%29
.
.
.
The difference of 60 would be,
x-y=60
y=x-60
So the product of the two numbers is,
P%5B2%5D=x%2Ay=x%28x-60%29
.
.
.
Yes, you can set up a problem however you want.
Sometimes it won't make sense.
Example: Make a shape using 40' of wire so that the enclosed shape has the minimum area. You could just make a straight line of 4 lengths of 10' of wire so that the enclosed area is zero.
Well if you're a farmer and you want to make a pen,
why would you want the minimum area,
you'd want the maximum area for your animals to graze.

Answer by MathTherapy(10552) About Me  (Show Source):
You can put this solution on YOUR website!

Why are these two similar problems set up differently? and why are they set up in that way at all?
Find two numbers whose difference is 60 and whose product is a minimum? Why does this use f(x) = x(x-60)
and
Find two numbers whose sum is 60 and whose product is a maximum? Why does this one use f(x) = x(60-x)
I fundamentally don't understand what is happening and how that function was created at all.
And will there ever be a case where it's a difference and the product is a maximum. Or where it's a sum and the product is a minimum. I don't understand what this question is asking.
So confused...
Please help!
1st problem:

Let larger number be x, and smaller, y

Then we have: x - y = 60______y = x - 60 ------- eq (i)

Also, y = xy ------- eq (ii)

y = x(x – 60) ------- Substituting x – 60 for y in eq (ii)

y+=+x%5E2+-+60x --------- eq (ii)

MINIMUM occurs at: x+=+-+b%2F%282a%29, or at: x+=+-+%28-+60%29%2F%282+%2A+1%29, or at x+=+60%2F2, or at: x = 30

With MINIMUM occurring at x = 30, we get: 

y = 30 – 60 ----------- Substituting 30 for x in eq (i)

y = - 30

As seen, the numbers that differ by 60, and have a MINIMUM product are:  highlight_green%28system%28-+30_and%2C30%29%29

MINIMUM: 30(- 30), or - 900



2nd problem:

Let larger number be x, and smaller, y

Then we have: x + y = 60_____y = 60 - x ------- eq (i)

Also, y = xy ------- eq (ii)

y = x(60 - x) ------ Substituting 60 – x for y in eq (ii)

y+=+60x+-+x%5E2

MAXIMUM occurs at: x+=+-+b%2F%282a%29, or at: x+=+%28-+60%29%2F%282+%2A+-+1%29, or at x+=+%28-+60%29%2F%28-+2%29, or at x = 30

With MAXIMUM occurring at x = 30, we get: 

y = 60 - 30

y = 30



As seen, the numbers that sum to 60, and that have a MAXIMUM product are the same, at: highlight_green%28system%2830_and%2C30%29%29

MAXIMUM: 30(30), or 900