Gina is standing near a building and notices that the angle...

Gina is standing near a building and notices that the angle of elevation to the top of the building is $68^{\circ} .$ She then walks 72 feet further away from the building and notices that the angle of elevation to the top of the building is now only $51^{\circ} .$ Find the height of the building. a. $25 \mathrm{ft}$ b. $56 \mathrm{ft}$ c. $149 \mathrm{ft}$ d. $180 \mathrm{ft}$

Gina is standing near a building and notices that the angle of elevation to the top of the building is $68^{\circ} .$ She then walks 72 feet further away from the building and notices that the angle of elevation to the top of the building is now only $51^{\circ} .$ Find the height of the building.
a. $25 \mathrm{ft}$
b. $56 \mathrm{ft}$
c. $149 \mathrm{ft}$
d. $180 \mathrm{ft}$

Key Concepts

Tangent Function

The tangent function in trigonometry represents the ratio of the length of the opposite side to the adjacent side in a right triangle. In height and distance problems, tangent is used to relate the height of an object to the horizontal distance from the object, given a particular angle of elevation.

Right Triangle Geometry

Right triangle geometry is a fundamental component of trigonometry, where one angle is 90 degrees. This concept underpins the use of trigonometric functions such as sine, cosine, and tangent to find unknown sides or angles, particularly in practical applications like measuring the height of a building.

Angle of Elevation

The angle of elevation is the angle formed between the horizontal line of sight and the line reaching up to an object. It is used in problems involving heights and distances to compute unknown measurements by applying trigonometric ratios in right triangles.

Systems of Equations

In many real-world trigonometry problems, multiple observations lead to the formation of a system of equations. By setting up equations based on different positions or observations and then solving them simultaneously, one can determine unknown values such as height or distance.

Transcript

00:01 Okay, so let's take a look at this.

00:02 Gina's standing near a building and notices that the angle of elevation to the top of the building is 68 degrees.

00:06 Then she walks 72 feet further away from the building and notice the angle to the top of the building is now only 51.

00:13 So basically here's gina standing here.

00:16 She moves 72 feet away over here.

00:18 So we basically have two right triangles.

00:21 We have the black one and the red one.

00:24 And both of those right triangles share the same exact height of the building.

00:30 So we're going to use our sohcahtoa, our trigonometry, to figure this out.

00:35 Now this one's a little tricky because you've got to set up a system of equations in order to solve for the height, or to solve for x and then solve for the height.

00:45 Because what we have right here is we really don't have any known sides.

00:49 We know that the distance she is from a building is something, and then later it's that x value plus 72 feet.

00:58 So we've got a couple of unknowns.

01:00 So i'm going to start with the first triangle.

01:01 If we're here, the height of the building is opposite from you, and this unknown distance is adjacent to you, so we'll use tangent.

01:10 So we'll set up our trig equation, tangent of 68 equals the height of the building over this distance right here, x.

01:18 Then we'll move gina over here and look at the red triangle.

01:22 Triangle so we are still using tangent so tangent of 51 equals the opposite side so the height over the adjacent side would be x plus 72.

01:35 So it looks like they both share this h so if we could solve for h in both equations and set this up and figure this all out we're going to be pretty good.

01:45 So what i'm going to do is i'm going to multiply by x on this side and this side to isolate the h.

01:54 And over here, i'm going to multiply by x plus 72 on both sides, x plus 72...

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