The angle of elevation to the top of a building changes from 20^∘ to 40^∘ as an observer advance

The angle of elevation to the top of a building changes from...

Key Concepts

Angle of Elevation

This concept refers to the angle formed between the horizontal line of sight of an observer and the line drawn from the observer's eye to a point of interest, in this case, the top of the building. It is a key measure in many real-world applications, particularly when determining heights or distances using trigonometric relationships.

Right Triangle Geometry

In problems like this, the situation is typically modeled using right triangles, where one vertex is at the observer’s position, the right angle is at the observer’s location on the horizontal ground, and the triangle’s opposite side represents the building’s height. This geometric setup allows the application of trigonometric functions to relate angles to side lengths.

Tangent Function

The tangent of an angle in a right triangle is the ratio of the length of the opposite side to the length of the adjacent side. This function is particularly useful for calculating unknown distances or heights when an angle of elevation is known, as it directly links the vertical height of an object to the horizontal distance from the observer.

Solving Trigonometric Equations

Setting up and solving trigonometric equations involves expressing relationships from different positions or conditions as equations and then using algebraic methods to find the unknown quantities. In the context of this problem, the different angles of elevation lead to two equations that can be equated or solved simultaneously to determine the building’s height.

Transcript

00:01 The question requires us to calculate the height of the building with the given dimensions.

00:08 I'm going to do a sketch of what is given in the question.

00:29 So the story is the person was observing first and he realized that the angle of elevation was 20 degrees.

00:41 Then he walked towards the building 75 feet and observed again this time the angle of elevation was 40 degrees so the question says calculate the height of the building so in order for you to calculate the height of the building you have to recognize it and then i'm going to make this remaining distance equal to x so that the whole adjacent side is complete.

01:16 Now we have got the opposite and the adjacent for both angles.

01:21 So in that case we are going to use turn.

01:25 So tan of an angle is equal to opposite over adjacent.

01:37 Now i'm going to start with 20 degrees.

01:43 So turn 20 degrees is equal to opposite, which is y, of adjacent.

01:54 The adjacent side is the whole adjacent of the triangle, which is 75 plus x.

02:04 Then what we are going to do is just cross multiply so that we create an equation without fractions.

02:16 Then we cross multiply 75 plus x times turn 20.

02:23 We get 75 1020 plus x turn 20 equals to y.

02:40 Then 75 turn 20 on the calculator gives us 27.

02:52 2028 and turn 20 when it's on it's 0 .364 x which is equal to y so we can call this one our equation 1 now we go to equation 2 so we just indicate that we are going to equation 2 there turn 40 degrees is equal to opposite and our opposite is y over adjacent or adjacent is x then we cross more apply we'll have x turn 40 equals to y then turn 40 is roughly about 0 .839 equals to y this is becomes our equation two.

04:10 Now we're just going to solve these two equations simultaneously so that we find y.

04:16 Our goal is to find y...

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