Prove mathematically that the lengths of the crank....

[maths85]

Homework Statement

Design an offset slider-crank mechanism which has the following specifications:
1. Stroke length of slider block C = 150 mm
2. Advance-to-return time ratio = 1.2
3. Offset height h = 50 mm

Prove mathematically that the lengths of the crank and of the connecting rod are AB
= 71 mm and BC = 178 mm respectively to the nearest millimetre.

I would appreciate any help to find the process and ecuations necessary to solve this problem.
I have tried to use differet ways to solve it but it gets very complicate and it suppose to be something easier..

I found a possible solution using the next representation and equations:

2. Homework Equations

We take into account that:

When I got this quadratic equations I realize that its getting too complicate, plus even solving this equations at the end there is not solution and I need to use a graphic way to find the length of the crank, and I have to use only mathematically process to solve it, I can't solve it graphically.
Another possible solution that I found was using the next steps:

The Attempt at a Solution

Another way that suppose to drive to calculate the lengths is using complex numbers but I started and it is quite complicate also.

Does anybody know where could I find a non very complicate way to solve this, my teacher told me that it suppose to be a mathematical method that is not very complicate?
I don't need anybody to make it for me, only a little help to where could I find formulas and the right approach to solve it.

Thank You.
Have a good day

 

[Nidum]

Back solve from the given answers .

 

[maths85]

Hello.
Iam still stuck, I am using the ratio formula to get the angles, I tried using the cosine and sin rules, but still not going anywhere, with these formulas that I put in the last part of my post I can't get anywhere because I don't have enough data, I need to use the length to go somewhere and that exactly what I need to calculate, so I can't use this data.
Any possible advice of how to proceed?
I guess I am missing something that I can´t realize