Find the angle AOC.

Let's analyze the image. We are given the angles \(\angle DOF = 30^{\circ}\) and \(\angle BOE = 40^{\circ}\). We need to find the angle \(\angle AOC\).

First, notice that \(\angle DOF\) and \(\angle BOE\) are vertical angles to \(\angle AOF\) and \(\angle COB\) respectively.

Vertical angles are equal. Therefore,
$$\angle AOF = \angle DOF = 30^{\circ}$$
$$\angle COB = \angle BOE = 40^{\circ}$$

Since \(\angle AOB\) is a straight angle (180 degrees), we have:
$$\angle AOB = \angle AOF + \angle FOC + \angle COB = 180^{\circ}$$

We can also write that \(\angle AOC + \angle COB = \angle AOB\), so
$$\angle AOC + 40^{\circ} = 180^{\circ}$$
From that, we express \(\angle FOC\) as
$$\angle AOC = 180^{\circ} - 40^{\circ} = 140^{\circ}$$

However, we need to express \(\angle AOC\) using \(\angle AOF\), \(\angle FOC\), \(\angle COB\). Since line OC divides \(\angle AOB\) into \(\angle AOC\) and \(\angle COB\) we can say that
$$\angle AOC = \angle AOB - \angle COB$$
\(\angle AOB = 180^{\circ}\), so:
$$\angle AOC = 180^{\circ} - 40^{\circ}$$
$$\angle AOC = 140^{\circ}$$

Another way to see this is to recognize that \(\angle DOC\) is also a straight line, so \(\angle DOC = 180^{\circ}\).
We also know that \(\angle DOF = 30^{\circ}\). Therefore:
$$\angle FOC = \angle DOC - \angle DOF$$
$$\angle FOC = 180^{\circ} - 30^{\circ} = 150^{\circ}$$
Also note that \(\angle AOF + \angle FOC = \angle AOC\). We found that \(\angle AOF = 30^{\circ}\), so
$$\angle AOC = 150^{\circ} - 30^{\circ} = 140^{\circ}$$

Thus, the angle \(\angle AOC\) is 140 degrees.

Answer: 140°