Thread: Semicircle & Chords (Urgent)

Hey guys. I have no idea how to even start this question. Any ideas?? Thanks!

"A semi-circle is dawn outwardly on a chord 'AB' of a circle with center 'O' and units raidus. The point 'C' on this semi-circle which sticks out of the given circle the furthest is on the radius 'ODC' which is perpendcular to 'A'. Of course the Length of 'OC' is dependant on the choice of chord 'AB' in the first place. Determine 'AB' so that 'OC' is maximum"

Hello, furnis1!

First, make a sketch . . .
[I'll use Calculus to maximize OC.]

A semi-circle is dawn outwardly on a chord of a circle with center and raidus 1.
The point on this semi-circle is the furthest from center
Of course, the length of is dependant on the choice of chord
Determine so that is maximum.

Code:

              * * *
          *           *  A
        *               o
       *              / :*   o
                   1/   :      o
      *           / θ  D: *
      *       O * - - - + * - - o C
      *           \     : *
                   1\   :      o
       *              \ :*  o
        *               o
          *           *  B
              * * *

The circle with center has radii

A semicircle is constructed (outward) with center and diameter
. . It has radii:

Let

In right triangle

Then: .


To maximize , set

We have: .

. . Hence: .

is maximum when


Then and is an isosceles right triangle.

Since its legs are length 1, its hypotenuse is: .

great answer prof! thanks for the help - that really cleared it up.