Question

A building casts a 15-foot shadow on the ground. A person standing at the end of the shadow can view the top of the building at a 60° angle with respect to the ground. Set up a proportion based on the similar triangles in this situation and the given special triangle. Show the proportion, and solve it to determine the exact height of the building.

Answer 1

To calculate the building's height with a 15-foot shadow and a 60° viewing angle, we use the properties of special right triangles (30-60-90) to set up a proportion and solve for h. The building's height is found to be approximately 25.98 feet.

Step-by-step explanation:

To determine the height of the building given a 15-foot shadow and a 60° viewing angle, we use the properties of special right triangles. Specifically, we can use a 30-60-90 triangle where the ratios of the sides are 1:√3:2. The building and its shadow form the two legs of a right triangle with the building as the opposite side to the 60° angle, and the shadow as the adjacent side.

For a 30-60-90 triangle, the length of the side opposite the 60° angle is √3 times the length of the side opposite the 30° angle. If we let h represent the height of the building, we can establish the following proportion: h / 15 = √3 / 1.

Solving the proportion for h gives us:

h = 15 √3
h ≈ 15 * 1.732
h ≈ 25.98 feet (exact height depends on the precision of √3 used)

Therefore, the height of the building would be approximately 25.98 feet.