Question

technology required. lada is visiting new york city to see the empire state building. she is 100 feet away when she spots it. to see the top, she has to look up at an angle of 86.1 degrees. how tall is the empire state building to the nearest foot?
type your answer in the box.
_____ feet

Explanation:

Step1: Identify the trigonometric relationship

We have a right triangle where the adjacent side to the angle of elevation (86.1 degrees) is 100 feet (the distance from Lada to the building), and we need to find the opposite side (the height of the building, let's call it \( h \)). The tangent function relates the opposite and adjacent sides in a right triangle: \( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \).

Step2: Substitute the known values into the tangent formula

Substituting \( \theta = 86.1^\circ \), adjacent = 100, and opposite = \( h \) into the formula: \( \tan(86.1^\circ) = \frac{h}{100} \).

Step3: Solve for \( h \)

Multiply both sides by 100 to isolate \( h \): \( h = 100 \times \tan(86.1^\circ) \).

Step4: Calculate the tangent of 86.1 degrees

Using a calculator, \( \tan(86.1^\circ) \approx 15.08 \) (make sure the calculator is in degree mode).

Step5: Compute the height

Multiply 100 by 15.08: \( h = 100 \times 15.08 = 1508 \) (rounded to the nearest foot).

Answer:

1508