Question

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°. 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.

\boxed{} meters

Explanation:

Step1: Identify the trigonometric relationship

We have a right triangle where the height of the rock is the opposite side (\(opp = 5\) m) to the angle of elevation of the sun (\(\theta = 31^\circ\)), and the distance from the rock (let's call it \(x\)) is the adjacent side. We use the tangent function, which is defined as \(\tan(\theta)=\frac{opp}{adj}\). So, \(\tan(31^\circ)=\frac{5}{x}\).

Step2: Solve for \(x\)

Rearranging the formula for \(x\), we get \(x = \frac{5}{\tan(31^\circ)}\). We know that \(\tan(31^\circ)\approx0.6009\) (using a calculator). Then, \(x=\frac{5}{0.6009}\approx8.32\).

Answer:

\(8.32\)