Statics Problem -- A bucket suspended between two walls by a rope and pulley

[Nicolas32]

Homework Statement

The pail and its content have a mass of 60 kg. If the cable BAC is 15 m long, determine the
distance y to the pulley at A for equilibrium. Neglect the size of the pulley.

Relevant Equations

Total F=0, Sine Law, Cosine Law

I approached this question with a free body diagram.
my Verticle length of BA is y-2
AC= 15-AB
Horizontal length of AC= a; Horizontal length of BA=10-a
I have too many unknowns in this problem and I don't know any directions(angle).
I know that I should start at the weight and try to resolve the AB and AC. But I don't know how the length relates with force.
Can someone just give me directions to approach this problem?
Thank you.

 

[hutchphd]

Draw a free body diagram for pulley A, showing the forces. This will elucidate the symmetry of the problem. The rest is simple geometry.
If you want more details you must show your work as per the homework policy.

 

[Nicolas32]

I tried finding the angle using sine rule(using only lengths),
(10 - y/tanθ)/sin∠BA(-x) = (15-y/sin∠CAX)/sin90 = (y-2)/ sin∠ AB(vertically downwards)

 

[Nicolas32]

Draw a free body diagram for pulley A, showing the forces. This will elucidate the symmetry of the problem. The rest is simple geometry.
If you want more details you must show your work as per the homework policy.

 

[haruspex]

View attachment 272249
View attachment 272250
I tried finding the angle using sine rule(using only lengths),
(10 - y/tanθ)/sin∠BA(-x) = (15-y/sin∠CAX)/sin90 = (y-2)/ sin∠ AB(vertically downwards)

Your notation is a bit hard to decipher, but you appear to be aware that AC and AB make the same angle to the horizontal.
Similar triangles?

 

[Nicolas32]

Your notation is a bit hard to decipher, but you appear to be aware that AC and AB make the same angle to the horizontal.
Similar triangles?

I was not aware of that. So if the item is in equilibrium, this causes the angles from the two strings to be equal to each other? With this info I can solve it. I'll be posting soon.

 

[haruspex]

I was not aware of that. So if the item is in equilibrium, this causes the angles from the two strings to be equal to each other? With this info I can solve it. I'll be posting soon.

This was what drawing the FBD per post #2 was supposed to show you.

 

[Nicolas32]

This was what drawing the FBD per post #2 was supposed to show you.

I see. My Y was 4.76m.
Thank you for your help.