Hello, Educated!

Your diagram didn't show up, but I'll take a guess.

There are 2 cogs, connected together by a belt.

Find the length of the belt.

The large cog has a radius of 11cm. The small cog has a radius of 4cm.

The distance between the centres of the 2 cogs is 25cm.

I got . $\displaystyle \angle AOB = 147.48^o$ and .$\displaystyle \angle YPZ = 212.52^o$

These happen to be correct.

In the diagram, consider quadrilateral $\displaystyle OPZB.$

. . Note that it has two right angles.

Code:

O *
| @ *
7 | * 25
| *
| *
Q * - - - - - - - - - * P
| 24 |
4 | | 4
| |
| |
B * - - - - - - - - - * Z
24

In right triangle $\displaystyle OQP\!:\;QP^2 + 7^2 \:=\:25^2 \quad\Rightarrow\quad QP = BZ = 24$

Also: .$\displaystyle \sin\theta \:=\:\frac{24}{25} \quad\Rightarrow\quad \theta \:\approx\:73.74^o $

and: .$\displaystyle \angle OPZ \:=\:180^o - \theta \:=\:106.26^o$

Can you finish up?