In order to estimate the height of a building, two students...

In order to estimate the height of a building, two students stand at a certain distance from the building at street level. From this point, they find the angle of elevation from the street to the top of the building to be 35°. They then move 250 feet closer to the building and find the angle of elevation to be 53°. Assuming that the street is level, estimate the height of the building to the nearest foot.

Key Concepts

Solving Systems of Equations

When multiple measurements or observations are taken from different positions, they often lead to a system of equations that describe the same physical situation. Solving these equations simultaneously allows the determination of unknown variables, such as the height of a building or the distance to it. This process involves applying algebraic techniques to intersect the conditions given by different trigonometric relationships.

Angle of Elevation

The angle of elevation is the angle formed between a horizontal line and the line of sight when an observer looks upward toward an object. It is a foundational concept in trigonometry that helps relate the height of an object to the distance from the observer, and it is commonly used in problems involving measuring the heights of inaccessible objects like buildings or mountains.

Tangent Function

The tangent function in trigonometry is defined as the ratio of the length of the side opposite an angle to the length of the adjacent side in a right triangle. This function is crucial when dealing with angles of elevation or depression because it directly relates the height of an object to the horizontal distance from its base.

Right Triangle Geometry

Right triangle geometry involves the properties and relationships intrinsic to triangles with one 90° angle. In problems involving angles of elevation, the scenario can be modeled using a right triangle where the building's height is the opposite side, and the horizontal distance from the building forms the adjacent side. Understanding these relationships is essential for setting up correct trigonometric equations.

Transcript

00:01 All right, so number 67 does not give us a picture, so i'm kind of going to draw the picture as i read the problem.

00:06 It says, in order to estimate the height of a building, two students stand at a certain distance from the building at street level.

00:12 Okay, so i'm going to say this right here.

00:14 That'll be the building.

00:16 And two students.

00:17 So here's one, here's the second one.

00:19 Okay.

00:22 From this point, they find the angle of elevation from the street to the top of the building to be 35.

00:27 Oh, i forgot.

00:28 Sorry, they're standing together.

00:30 Okay, so the students are right here, and they find that the angle of elevation, to the top of the building is 35.

00:38 I'm going to put that right here.

00:40 35 degrees.

00:46 Okay.

00:47 Then they move 250 feet closer to the building.

00:51 So they've moved, let's say, right about here.

00:56 So this distance, right, that i'm highlighting, that is, represents the 250 feet that they moved.

01:03 And from that point, they find the angle of elevation to be 53.

01:07 So looking up at the level of elevation, the building this angle is 53.

01:15 Assuming that the street is level, estimate the height of the building to the nearest foot.

01:19 So i'll put an x right here that represents this part of the triangle.

01:23 That is the height of the building.

01:28 So we're actually going to have to find two pieces here.

01:34 I can use the fact that i know this angle here is 53 and that will help me determine this length...

Please give Ace some feedback