Please help people smarter than me

[Mathematical1928]

I started solving this for fun but have no idea what to do. PLEASE HELP and if you can show how you did it so I can learn. Solving for R.

 

[lev888]

We know what the value of A₂ is.
A₂ is also a sum of areas of several shapes. Can you write the expressions for those areas?
This will give you an equation.

 

[Mathematical1928]

position of the arch on the Y axis is X = 40 - V
A2 = (((X*10) /2)*2) + (400 - (((X*10) /2)*2))

 

[Mathematical1928]

position of the arch on the Y axis is X = 40 - V
A2 = (((X*10) /2)*2) + (400 - (((X*10) /2)*2))

10X +(400 - 10X) = 400 = A2

 

[lev888]

10X +(400 - 10X) = 400 = A2

Sorry, I don't follow.

We have a rectangle. It's divided into 2 shapes by the arc. We are told that these 2 shapes have equal areas (A1 = A2). Do we know the area of the rectangle? If we do, can we calculate A1 and A2?

 

[Mathematical1928]

Area of the rectangle is 800 = 40 * 20
A1 = A2 = 400
Area of the Triangles is (10*X)/2
X Being the position of the points of contact of the the Arch on the Y axis

 

[lev888]

Ok. What about the other shape? What would be the resulting equation?

 

[Dr.Peterson]

You'll need to find the angle of the arc, and find the area of the sector plus the area of the two triangles. Then see this equal to half the area of the rectangle.

Typically in this kind of puzzle, you end up with a transcendental equation that can't be solved algebraically; you'll need some sort of numerical solution.

 

[pka]

I have two wishes for this thread:
1) that the image were better drawn.
2) that the problem were completely stated with all details.
Look at this web page. The circular segment in the OP is part of area \(\displaystyle A_2\).
In fact, looking at the webpage we know that cord length \(\displaystyle c=20\).
A better labeled drawing would enable us to discuss this question.

 

[Jagulba]

This seems not to be that easy (linear).. the best formula that i got have arcsin(10/R) and R^2 which... I’m not sure if you can easily find a solution
A hint would be to transfer arcsin to power of R.... even then it’s not easy
I don’t want to give the number but its close to 20...