Problem 60 - Algebra and Trigonometry

Question

After a wind storm, you notice that your 16 -foot flagpole may be leaning, but you are not sure. From a point on the ground 15 feet from the base of the flagpole, you find that the angle of elevation to the top is \(48^{\circ} .\) Is the flagpole leaning? If so, find the acute angle, to the nearest degree, that the flagpole makes with the ground.

Step-by-Step Solution

Answer

Calculate \( \tan(48^{\circ}) \) and compare it to \( \frac{16}{15} \). If the two values are equal, the pole is not leaning. If not, the pole is leaning and the acute angle \(\alpha\) that the pole makes with the ground is \( \alpha = \arctan(\frac{16}{15}) \) rounded to the nearest degree.

Step 1: Identifying Given Information

The given information is that the flagpole is originally 16 ft tall and 15ft from a measuring point on the ground. The angle of elevation from that point to the top of the flagpole is \(48^{\circ}\).

Step 2: Creating A Diagram

Consider the situation as a right triangle. The bases of the triangle is the distance from the measuring point to the flagpole, which is 15 feet. The height of the triangle is the height of the flagpole, which is 16 feet.

Step 3: Applying Trigonometric Relations

The tangent of the angle of elevation is equal to the height/length ratio, i.e., \( \tan(48^{\circ}) = \frac{16}{15} \). Calculate \( \tan(48^{\circ}) \) and compare it to \( \frac{16}{15} \).

Step 4: Drawing Conclusion

If \( \tan(48^{\circ}) \) is equal to \( \frac{16}{15} \), then the flagpole is not leaning. If \( \tan(48^{\circ}) \) is not equal to \( \frac{16}{15} \), then the flagpole is leaning. In that case, find the acute angle \(\alpha\) that the flagpole makes with the ground by calculating \( \alpha = \arctan(\frac{16}{15}) \) and round to the nearest degree.