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Drawing vertical line through a circle to find for what x does that vertical line div

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Hi all, I would appreciate some help with the problem:

The problem state: "A building design consultant is going to make a 14 foot diameter windowwith a vertical cut through the glass so that 2/3 of the glass can be tinted. Wheres should this cut be made?"

As you can see in the above attachment, I was able to conclude into some sort of equation, but when I solve that equation to 49*pi/3 via wolfmath, I got about 6.83 = x, which does not make sense because the radius is 7 and that wouldn't be 1/3 if you draw the vertical line through it to divide the circle.

If you could help me out, I would really appreciate it.

Thank you,

Re: Drawing vertical line through a circle to find for what x does that vertical line

First note that the area of this circle would be

If we place the circle on a set of axes centred at the origin, it has equation then draw in the vertical line somewhere to the right of the y-axis, we wish to know what value of x this is so that it splits the circle into areas in the ratio 2:1. So that means the area bounded between this line and the circle on the right will be , and since this is evenly distributed above and below the x-axis, the area above the x-axis is .

So we can set up an integral for the area bounded by the circle, the x-axis and the vertical line as .

Equating this to the known area we have .

In order to evaluate this integral and solve for the value of a, we need to make a trigonometric substitution and note that when and when . Substituting gives

Unfortunately there is no way to evaluate the value of a here exactly, but you should be able to get a numerical answer using a CAS :)

Re: Drawing vertical line through a circle to find for what x does that vertical line

:) Brilliant! Thank you so much.

Re: Drawing vertical line through a circle to find for what x does that vertical line

If the ends of a chord drawn in a circle radius subtend an angle at the centre of the circle, then the area 'cut off' by the chord is

That means we need or,

That requires a numerical solution, which turns out to be

For a circle of radius it follows that the centre of the chord should be at a distance of (approx) from the centre of the circle.

Re: Drawing vertical line through a circle to find for what x does that vertical line

Very well done in fact i was stuck at the same stage. i used the formula ∫▒√(a^2- x^2 ) dx= x/2 √(a^2- x^2 )+ 1/2 sin^(-1)〖x/a〗+c directly the issue remains how to get a ??

Re: Drawing vertical line through a circle to find for what x does that vertical line

You need to use a numerical method and/or technology such as a CAS. There is no way to isolate a exactly when it's both inside and outside of a transcendental function (unless it's a very special case where a solution sticks out at you).