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Problem

I'm stuck with the above problem (see image). I understand, i think, that i need to find the maximum height of the triangle using line AB as its base. To do that i need to find a function for the height, differentiate it, and then solve for when the differential is equal to 0 (max distance from AB).

Apparently the answer is 6 - sqrt(3) feet. I can't figure out how to get that.

I've included another image of my initial approach, but am getting stuck because this seems like a multivariable problem where the height is dependent on both the angle,theta, and x. i'm not studying multivariable calculus so i must be missing something. Really appreciate the help! initial thought

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2 Answers 2

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It seems the answer to be purely geometric. The problem is symmetric. With this and using your notations, 2x+2y=L and h^2=y^2-x^2, where L=8 ft. Physics has nothing to do with it.

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If pulley C is in a position level with pulley B then the weight will hang 4ft directly below B.

If the pulleys are frictionless then the system will stabilise as an isosceles triangle with 'base' AB and sides AC and BC each of 3ft length ... which gives the vertical distance as the x in a 1,x,3 right-angled triangle, which is sqrt(8), or 2sqrt(2).

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