Question

archimedes went to sleep beside a big rock. he wanted to get up at 8 am, but the alarm clock was yet to be invented! he decided to sleep at the spot where the rocks shadow should end when its 8 am so as to be awakened by the direct sunlight.

archimedes knew that at 8 am, the sunlight reaches the ground at an angle of (43^circ). the rock beside which he slept was 7 meters tall.

how far from the rock did archimedes go to sleep?
round your final answer to the nearest hundredth.

(square) meters

Explanation:

Step1: Identify the trigonometric relationship

We have a right triangle where the height of the rock is the opposite side (\(opp = 7\) m) to the angle of elevation (\(\theta = 43^\circ\)), 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, \(\tan(43^\circ)=\frac{7}{x}\).

Step2: Solve for \(x\)

Rearranging the formula for \(x\), we get \(x = \frac{7}{\tan(43^\circ)}\). We know that \(\tan(43^\circ)\approx0.9325\) (using a calculator). Then, \(x=\frac{7}{0.9325}\approx7.51\).

Answer:

\(7.51\)