Find the height of the building across the street

Question

A woman entering an outside glass elevator on the ground floor of a hotel glances up to the top of the building across the street and notices that the angle of elevation is 49°. She rides the elevator up three floors (60 feet) and finds that the angle of elevation to the top of the building across the street is 32°. How tall is the building across the street? (Round to the nearest foot.)

Answer 1

Draw a figure!

Let y be the height from the 3rd floor to the top of the building across the street and

let x be the distance across the street.

From your figure you should see that y/x = tan 32 and

(y+60)/x = tan 49 so that

y/tan 32 = (y+60)/tan 49 then

y(tan 49 - tan 32) = 60 tan 32

I will leave the arithmetic to you.

Answer 2

Draw a picture with the two buildings and mark off points at 0 and 60 ft. on one and a point at the top of the other and show the two angles of elevation.

You will see that there is a triangle made up of the three points, with the length of one side 60 ft', and the angle of elevations make two of the angles of the triangle 90° - 49° and 90° + 32°. The third angle will then be 180° - (90°-49° + 90° + 32°) = 17° The 17° angle will be opposite the 60 ft. side, and the 90° + 32° = 122° angle will be opposite the bottom side of the triangle. We wish to solve for this side, as the height of the other building, while not the length of one of the sides of the triangle, can be solved for once we know the length of the bottom side, as it will be the hypotenuse of a right triangle where the two sides have length of the distance between the buildings and the angles are 90°, 49°, and 41°, with the height of the building the side opposite the 49° angle and thus have length hypotenuse * sin(49°).

For the length of the bottom side, sin 17°/60 = sin 122°/x

x = 60 sin 122°/sin 17°

Then, the height of the building will be x sin 49° = 60 sin 122° sin 49°/sin 17° = 131.3458

Answer 3

Starting with the rt triangle: α=41°

β=49° (given)

γ=90°

The building is side "b"; ground is side "a": side "c" is the hypotenuse. .

Remember this formula; it only works with rt triangles: a^+b^2=c^2.

You want side "b".

Before we go any further we've got to add an isosceles into the picture.

Here β=41°(90-49)

γ=122° (90+32)

α=17°

a=60' (given)

Another formula to remember is: a/sin=b/sinβ=c/sinγ (any triangle).

To get the building we first need side "c" of the rt triangle; which is also side "c" of the isos.

So: 60/sin17=c/sin122

60/.292=205.21*.848=174.03'="c"

To get the height b/sin49=174.03 (sin 90°=1); so 174*.754=131.34582......'