Archimedes went to sleep beside a big rock. He wanted to get up at $7$ AM, but the alarm clock was yet to be invented! He decided to sleep at the spot where the rock's shadow should end when it's $7$ AM so as to be awakened by the direct sunlight. Archimedes knew that at $7$ AM, the sunlight reaches the ground at an angle of $31^\circ$. The rock beside which he slept was $5$ meters tall. **How far from the rock did Archimedes go to sleep?** *Round your final answer to the nearest hundredth.* [[☃ math-keypad 1]] meters