Question

1. a hiker is standing 220 feet from the base of a hill. the angle of elevation from where he is standing to the top of the hill is 29°. how tall is the hill?
2. a 30 foot ladder forms an angle of 50° with the ground when place against a brick wall. how far up the wall will the ladder reach? how far away from the base of the wall is the ladder?
3. alex is standing on the ground and looks up to see a plane flying in the sky. if it is flying at an altitude of 2 miles and the distance along the ground is 5 miles to a point directly below the plane, what is the angle of elevation that alex has to look up?
4. michael, whose eyes are five feet off of the ground, is standing 30 feet away from the base of a build and looking up at a 50° angle of elevation to point on the edge of buildings roof. to the nearest foot, how tall is the building?

Explanation:

Step1: Set up the trigonometric relationship

Let \( h \) be the height from Michael's eyes to the point on the roof. We know the adjacent side \( x = 30\) feet and the angle of elevation \(\theta=50^{\circ}\). Using the tangent function \( \tan\theta=\frac{h}{x}\), so \( h = x\tan\theta\).

Step2: Calculate \( h \)

Substitute \( x = 30\) and \(\theta = 50^{\circ}\) into the formula. \( h=30\times\tan(50^{\circ})\). Since \(\tan(50^{\circ})\approx1.191753592\), then \( h = 30\times1.191753592\approx35.7526\) feet.

Step3: Find the total height of the building

The total height \( H\) of the building is the height from the ground to Michael's eyes (\(5\) feet) plus \( h\). So \( H=5 + h\). Substitute \( h\approx35.7526\) into the formula, \( H\approx5+35.7526 = 40.7526\approx41\) feet.

Answer:

The building is approximately \(41\) feet tall.