Question

a building 73.45 feet tall has a shadow that is 76.28 feet long. find the angle of elevation of the sun to the nearest hundredth of a degree. the angle of elevation is \\( \square \\) degrees. (round to the nearest hundredth as needed.)

Explanation:

Step1: Identify the trigonometric relationship

We can model this situation as a right triangle, where the height of the building is the opposite side (\(opp = 73.45\) ft) to the angle of elevation, and the length of the shadow is the adjacent side (\(adj = 76.28\) ft) to the angle of elevation. The tangent of an angle \(\theta\) in a right triangle is given by \(\tan(\theta)=\frac{opp}{adj}\). So we have \(\tan(\theta)=\frac{73.45}{76.28}\).

Step2: Calculate the ratio

First, we calculate the value of \(\frac{73.45}{76.28}\). Let's compute that: \(\frac{73.45}{76.28}\approx0.9629\).

Step3: Find the angle

To find the angle \(\theta\), we take the arctangent (inverse tangent) of the ratio we just calculated. So \(\theta=\arctan(0.9629)\). Using a calculator (making sure it's in degree mode), we find that \(\arctan(0.9629)\approx43.93^{\circ}\).

Answer:

\(43.93\)