You measure angle of elevation from ground to top of building as 32°. When move 50 m closer to building, angle of elevation is 53°. How high is building?

Question

alegebra problem, involves sin, cos, tan, csc, sec, cot

Answer 1

Hi Makina-

Here's how you can solve this problem.  Let's start with a drawing:

 

                             |

                             |

                             |

                             |

                             | 

                             | 

(32^o)        (53^o)       | 

X-  (50m) -Y    (Zm) |

 

The first thing to recognize is that this creates two right triangles.  When we have right triangles, we can use the trigonometric functions to figure out the missing sides.

 

First, we are trying to find the height, and we only know the length of the other base.  This tells us that we will likely be using the tan function.  First, we can say the following:

 

               height

tan(32) = --------

               50 + z

 

This is problematic because we have two variables to solve for.  Before tackling the above equation, we note that we can also create a second equation:

 

               height

tan(53) = --------

                  z

 

What you should notice is that you can now use substitution to answer this question.  For example, if you isolate z in the second equation, you get:

 

      height

z = ---------

      tan(53)    

 

Then sub this into the first equation:

 

                       height
tan(32) = ------------------------
               50 + height/tan(53)

 

                  height

.625 =  -------------------

           50 + height/1.327

 

31.25 + .625(height)/1.327 = height

31.25 + .471(height) = height

31.25 = .529(height)

height = 59.07 meters

Answer 2

You measure angle of elevation from ground to top of building as 32°.

When you move 50 m closer to building, angle of elevation is 53°.

How high is building?

=====

h = height of building

x = distance from building when 53° elevation

tan(53°) = h/x ==> x = h/tan(53°)

tan(32°) = h/(x+50m) ==> x = h/tan(32°) - 50m

h/tan(53°) = h/tan(32°) - 50m

h/tan(32°) - h/tan(53°) = 50m

h(1/tan(32°) - 1/tan(53°)) = 50m

h = 50m / (1/tan(32°) - 1/tan(53°))

h ≈ 59.0471807553865m

Answer 3

Let's plot 4 points.

 

Draw a building

Point A is the location (on the ground) of the 1st spot you found the 1st measurement (the 32º angle)

Point B is the location (on the ground closer to the building than the 1st spot) of the 2nd spot you found the         2nd measurement (the 53º angle)

Point C is the location at the top of the building.

Connect A to B, A to C, and B to C - making an obtuse triangle

Label AB - 50m

Label CD - H (represents Height of Building - what you are trying to find out)

 

NOW, extend AB to where it meets the building.

Label the point where AB extended meets the building D

Label BD - X (we don't know how far this is --- yet)

 

 

 

If you connect the 'dots', you should have:

 

 

                                                                                                                • C (top of building)

 

 

 

                                                                                                                   h -- height of building

 

 

 

 

•A (32º) -----------50m---------------- • B (53º)---------- x ------------ • D (base of building)     

 

 

We first need to solve for x.

 

Using the rules of tangents -opposite over adjacent- and using the base of the building as a right angle (unless you're in Pisa), we have:

 

tan(32º)=    h            AND           tan(53º)=    h   

                (50+x)                                                x

 

 

 

Solving for h for both equations, we have:

 

h = tan(32º)(50+x)           AND         h = x tan(53º)

 

Using subitution [sic], we have:

 

tan(32º)(50+x) = x tan(53º)

 

Solve for x.   Once you find x, use either equation for h (I'd suggest h = x tan(53º) )