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 degree. the rock beside which he slept was 5 meters tall.

8.3 meters I am going to assume that Archimedes is capable of determining the horizontal distance from the peak of the rock to the ground. So we have a right triangle, with and angle of 31 degrees and the opposite side being 5 meters tall. So we want the adjacent side (the horizontal distance). This screams "tangent" which is defined as opposite over adjacent. So 5/x = tan(31) 5 = x*tan(31) 5/tan(31) = x 5/0.600860619 = x 8.321397412 = x So Archimedes needs to place his head 8.3 meters away from the point directly underneath the peak of the rock. Too bad for him that not only hasn't the alarm clock been invented, but meter itself won't be developed for about another 2000 years either.