Thread: Optimization World Problem?!

The sum of two nonnegative numbers is 20. Find the numbers if

a) the sum of their squares is as large as possible; as small as possible.

b) one number plus the square root of the other is as large as possible; as small as possible.

I'm not good at word problem, so any help would be great.

Does this topic relate to differensiation? Mean this question is in differensition chapter?

if yes then you have to use maxima and minima concept.

How?

if you don't know then i will write in next post..

Thanks

Originally Posted by Seigi

I'm not good at word problem, so any help would be great.

a)

It's given the sum of two non-neg numbers is 20, and the sum of their squares is the maximum/minimum.

(1) x + y = 20
(2) x^2 + y^2 = M(inimum)

We can write eqn (1) in terms of x (or y, it doesn't matter... whichever is easier but in this case it's the same)

so
(3) x=20-y

We plug eqn (3) into eqn (2):

M = (20-y)^2 + y^2
Now we differentiate....
M' = -40 + 4y
Now we solve the right hand side for zeroes, and it's obvious y=10

If y=10, then x must be 10 for the two numbers to add up to 20, per the condition.

So, x=10 y=10 is the sum of the square which is the minimum (smallest as possible).

There isn't really a "as large as possible" since the graph only has a minimum and not a maximum. The "maximum" would be something like x=19.99999999.... y=0.00000001 and we can make it as large as we like by repeating the .9999 and reflecting it on the x to make it equal to 20.



You do the same thing to solve your next question. And the answer should be 19.75 and 0.25. If you are unsure how to proceed and how to arrive to the answer, then let me know