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Calculus/Maximum-minimum problem

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Hi Paul,

I have a word problem regarding optimization that I don't quite get and know where to start and I can't find in this
http://en.allexperts.com/q/Calculus-2063/2009/11/Maximum-minimum-problem-41.htm column.

It is.

How close does the circle with radius sqrt(2) and center (2,2) come to the origin?

Thanks,
Sam

Answer
Distance
Distance  
Questioner:am
Country:British Columbia, Canada
Category:Calculus
Private:No
Subject:Optimization (local/global minima)

Question:
Hi Paul,

I have a word problem regarding optimization that I don't quite get and know where to start and I can't find in this
http://en.allexperts.com/q/Calculus-2063/2009/11/Maximum-minimum-problem-41.htm column.

It is.

How close does the circle with radius sqrt(2) and center (2,2) come to the origin?

Thanks,
Sam
...................................
Yes, I know -- it's not there.  What IS there is a classically elegant and detailed explanation of how to do M-M problems, if I may say so.

On the diagram, (which you always start with), I flipped the circle and point.  The result should be the same.

[Note: the distance from a point, such as (2,2), to some set of points, such as the points on the circle, is the minimum of all such distances.

Variables:  the x- and y-coordinates of P, on the circle.
To be minimized: Distance from P to (2,2).

So you want to:

minimize D = sqrt((x-2)^2 + (y-2)^2)

subject to the constraint:

x^2 + y^2 = 2

Let's go to work:

Minimize D^2 = (x-2)^2 + (y-2)^2

{This trick IS TO BE FOUND in that file of M-M problems}

D^2 = (x-2)^2 + (y-2)^2

D^2 = x^2 - 4x + 4 + y^2 - 4y + 4

y^2 = 2 - x^2,

y = sqrt(2 - x^2)

D^2 = x^2 - 4x + 4 + 2 - x^2 - 4 sqrt(2 - x^2) + 4

D^2 = - 4x - 4 sqrt(2 - x^2) + 10

Now you should be able to differentiate, etc...

===============================================

Suggestion:  Write x,y both as functions of theta, the central angle.

Then P is P(s2 cos t, s2 sin t)

[s2 is shorthand for square root of 2]

D^2 = (2- s2 sin t)^2 + (2 - s2 cos t)^2

and the work is quite easy after that.  

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Paul Klarreich

Expertise

All topics in first-year calculus including infinite series, max-min and related rate problems. Also trigonometry and complex numbers, theory of equations, exponential and logarithmic functions. I can also try (but not guarantee) to answer questions on Analysis -- sequences, limits, continuity.

Experience

I taught all mathematics subjects from elementary algebra to differential equations at a two-year college in New York City for 25 years.

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(See above.)

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