A spot light `S` rotates in a horizontal plane with a constant angular velocity of `0.1 rad//s`. The spot of light `P` move along the wall at a disatnce `3 m`. What is the velocity of the spot `P` when `theta = 45^(@)` ?

Explanation

To solve the problem, we need to find the velocity of the spot of light \( P \) on the wall when the angle \( \theta \) is \( 45^\circ \). Here’s a step-by-step breakdown of the solution:

### Step 1: Understand the Geometry
We have a spotlight \( S \) rotating with a constant angular velocity \( \omega = 0.1 \, \text{rad/s} \) at a distance \( OQ = 3 \, \text{m} \) from the wall. The angle \( \theta \) is the angle between the line from the spotlight to the wall and the vertical line from the wall to the spotlight.

### Step 2: Relate the Angle to the Distance
Using the tangent function in the right triangle \( OQP \):
\[
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