Question

a building 34.48 feet tall has a shadow that is 39.99 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 (\(opp = 34.48\) ft) and the length of the shadow is the adjacent side (\(adj = 39.99\) ft) to the angle of elevation (\(\theta\)). The tangent function relates the opposite and adjacent sides: \(\tan(\theta)=\frac{opp}{adj}\).

Step2: Calculate the tangent ratio

Substitute the values: \(\tan(\theta)=\frac{34.48}{39.99}\approx0.8622\).

Step3: Find the angle using arctangent

To find \(\theta\), we take the arctangent (inverse tangent) of the ratio: \(\theta=\arctan(0.8622)\). Using a calculator in degree mode, we get \(\theta\approx40.89^\circ\).

Answer:

\(40.89\)