The sun is at an angle of elevation of \( 58^{\circ} \). A tree casts a shadow 20 meters long on the ground. How tall is the tree? \( \begin{array}{ll}\text { a) } 71 \mathrm{~m} & \text { go to station } 1 \\ \text { b) } 11 \mathrm{~m} & \text { go to station } 4 \\ \text { c) } 17 \mathrm{~m} & \text { go to station } 2 \\ \text { d) } 80 \mathrm{~m} & \text { go to station } 9 \\ \text { e) } 32 \mathrm{~m} & \text { go to station } 7\end{array} \)

To find the height of the tree, we can use the tangent of the angle of elevation. The formula we can apply is: \[ \tan(\text{angle}) = \frac{\text{opposite}}{\text{adjacent}} \] Here, the “opposite” side is the height of the tree (let's call it \( h \)), and the “adjacent” side is the length of the shadow (20 meters). Thus, we can set up the equation: \[ \tan(58^{\circ}) = \frac{h}{20} \] Calculating \( h \), we have: \[ h = 20 \tan(58^{\circ}) \approx 20 \cdot 1.6003 \approx 32.01 \text{ meters} \] So, the tree is approximately 32 meters tall. Did you know that the mathematics behind shadows has been around since ancient times? The Greeks, including the famous philosopher and mathematician Euclid, studied similar triangles and proportions, which are foundational to our understanding of geometry today! In practical situations like measuring a tree's height using shadows, a common mistake is to confuse the angles. It's crucial to ensure you're measuring the angle of elevation correctly and not inadvertently using the angle of depression. Always double-check your angles, and don’t forget to use a protractor for precise measurements!