Optimisation question #1

[Win_odd Dhamnekar]

Optimisation question #1 final answer

It looks like you've skipped ahead.

We don't know side a, so the expression above won't give us the height. We're given a circle with radius r, so that's the only value that we "know". In order to write a height function h(r), we need to express the height in terms of r. Can you do this?

Also, it's not the triangle's height that minimizes its perimeter. You discovered, by finding the critical value \(\displaystyle \pi\)/6, that the triangle's perimeter is minimized when sides a and b are chosen to be equal. In other words, the (circumscribed) isosceles triangle's perimeter is minimized when it's an equilateral triangle.

I would refer back to the diagram. Two line segments comprise the height. We now have expressions for each of them, in terms of r and x. (You found the value of x that minimizes the perimeter; use it).

Hello,So height of the circumscribed equilateral triangle must be 3r so that we get its perimeter minimum.

 

[mmm4444bot]

h(r) = 3∙r

Somebody gives you a circle's radius, and they ask for the height of the circumscribing isosceles triangle having minimum perimeter, and you tell them that height is three times the radius.

Congratulations!

 

[Win_odd Dhamnekar]

Thanks

h(r) = 3∙r

Somebody gives you a circle's radius, and they ask for the height of the circumscribed isosceles triangle that has the smallest perimeter, and you tell them that height is three times the radius.

Congratulations!

Hello,
Many many thanks for guiding me in my journey to arrive at correct answer.

 

[mmm4444bot]

You're welcome. You've also discovered that, when you have an inscribed circle and equilateral triangle, the circle's radius is located from the base at one-third the triangle's height. (Handy) :cool:

If you like these types of geometry setups with trig, you might be interested in this pre-calculus exercise (stop the video at [02:11] to see what you think of the puzzle, before continuing). Maybe you get ideas for constructing objects in calculus scenarios.