Question

need help with an optimization problem
CIrcle in a triangle

Answer 1

What are the radius and area of the circle of maximum area that can be inscribed in an isosceles triangle whose two equal sides have length 1?

Answer 2

area of a circle \[A=\pi r^2\]
Area of a triangle \[A=(1/2)bh\]

Answer 3

\[\frac{dA}{dt}=2 \pi \frac{dr}{dt}\]

Answer 4

@AccessDenied any thoughts?

Answer 5

well, my first reaction to the problem was, "oh wow, this seems like a beautiful problem using different properties of triangles! :D"

including the incircle, which is the point of concurrency of the angle bisectors, and also the center of the largest circle inscribed in the triangle

Answer 6

yes, perhaps but it is a calculus problem - must solve by optimization problem techniques (max and min)

Answer 7

yes, it gets interesting when considering that we are given the fact that the legs of the isosceles triangle are 1, which leaves the base to vary. how the base varies also changes the incircle's area.

Answer 8

I think I also, need to take the derivative of the other
\[\frac{dA}{dt}=h \frac{db}{dt}+b \frac{dh}{dt}\]

Answer 9

well I am stuck here, I guess I better go read some more of my calculus books.

Answer 10

i'll see if i can think of how exactly to approach this problem in the mean time -- its not coming to me right now...

Answer 11

I know sometimes talking about it helps. One of Cornell's R - Recite

Answer 12

i think the key relations in this problem are between the area with respect to the radius, and the radius with respect to the base we choose, and the restriction is that the base must allow our triangle to, well, actually be a triangle. that would yield...

1 + 1 > b
2 > b, and
1 + b > 1
b > 0

Answer 13

the next step would be relating the base to the radius of the incircle, which i imagine entails some magic, coordinate geometry modelling, or something i have not learned that would probably make the problem go along smoothly

Answer 14

(although, i cannot guarantee that this is how the problem is supposed to be done, i am also unsure, but im just going with what i know about this stuff!)

Answer 15

I believe I have two equations - one is primary and the other secondary.
I am suppose to sub the secondary equation into the primary
I am referring to the equations above where I tool the derivative of

Answer 16

well I am going to go look at some more calculus books.......
Thanks :)

Answer 17

@AccessDenied Are you still logged on?

Answer 18

ah, i am now

Answer 19

sorry, i got dc'd earlier, and all my notification links disappeared

Answer 20

no don't worry. I had posted a ppt earlier - some old classnotes - were you able to view it

Answer 21

yeah, i didnt look through it completely yet -- i was actually trying to model this using coordinate geometry and am still working it out.. it seems like im doing an absurd amount of work for an optimization problem tho, so meh. :P

Answer 22

Here is what I remember about optimization

Answer 23

the 'primary equation' would be the area of the circle -- its what we're maximizing...
the radius would depend on the base. I'm uncertain if my coordinate model actually is yielding the correct answer, but i did find a relation between the radius and the base.

r = sqrt(1 - (b^2)/4) / 2

Answer 24

I can't see straight anymore, maybe I will look at this problem again tomorrow....
thanks for all of your help :) You have been very kind with your time and knowledge

Answer 25

i dont think my coordinate model + explanation will help... lol
its about a page long of just finding equations of things and then setting equal to other things. ;p

Answer 26

I am sure once I figure it out, it will not be so bad.... I sometimes make the process more complicated than it needs to be
Talk to you tomorrow....

Answer 27

mm... i feel i do that with these coordinate models all the time...
well, good luck in working this one out -- its definitely harder than previous optimization problems i remember. :P

goodbye!

Answer 28

@AccessDenied are you available? I want to share additional information

Answer 29

yes

Answer 30

I found this on line http://www.ajdesigner.com/phptriangle/equilateral_triangle_inscribed_circle_radius_r.php

Answer 31

interesting

Answer 32

I can sub this r into A=pi * r^2
before I take the derivative

Answer 33

of course a = 1 and b is my base