Question

1. length the sun is 25° above the horizon. find the length of a shadow cast by a building that is 100 feet tall (see figure).

Explanation:

Step1: Set up the tangent function

Let the length of the shadow be \(x\). In a right - triangle (building height, shadow length, and the line from the top of the building to the end of the shadow), \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\). Here, \(\theta = 25^{\circ}\), the opposite side is the height of the building (\(h = 100\) ft), and the adjacent side is the length of the shadow (\(x\)). So, \(\tan(25^{\circ})=\frac{100}{x}\).

Step2: Solve for \(x\)

Rearrange the equation \(\tan(25^{\circ})=\frac{100}{x}\) to get \(x=\frac{100}{\tan(25^{\circ})}\).
We know that \(\tan(25^{\circ})\approx0.4663\). Then \(x=\frac{100}{0.4663}\approx214.45\) ft.

Answer:

The length of the shadow is approximately \(214.45\) feet.