A woman standing on a hill sees a flagpole that she knows is 40 ft tall. The angle of depression to the bottom of the pole is 14°, and the a

Anonymous...this problem is quite a bit more difficult than the previous ones you have submitted.....I'll explain it...if you have trouble...let me know...it requires some "substitutions"

First.....as the woman looks at the pole...let's call the the opposite side to the angle of depression, "x"......this "x" represents only a part of the pole height that is below her sight line....so we have

tan(14) = x/d   where "d" is the distance the woman is from the pole ...this means that  

d = x/tan(14)

Now.........the part of the pole that is above her sight line is just 40-x...so using the angle of elevation and the tangent again, we have

tan(18) = (40-x)/d

Now....because  I want an equation in one variable, I'm going to substitute x/tan(14) for "d"in this second equation...so we have

tan(18) = (40-x)/ (x / tan(14))      ...  multiply both sides by (x/tan(14))..this gives

tan(18) (x / tan(14) = (40-x)         .... doing a little "rearranging" we have

[tan (18) / tan (14)] x   = 40- x      ......divide tan(18) by tan(14) = 1.303

1.303x = 40 -x                              .......add x to both sides

2.303x = 40                                  ........divide both sides by 2.303

x = 40/2.303 = 17.37                     this "x" represents the height of the pole below her sight line........but that's not what we're after...we want "d"...so using d = x/tan(14), we have

d = 17.37 / tan(14) = 69.7 ft.......and that's her distance from the pole

I realize you might get "stuck" on this one.......let me know....

I see a way that I could have made this easier........let's go from this step......

tan(14) = x/d   ........let's use this instead........x = d*tan(14)

And in the next part I had

tan(18) = (40-x) / d             let's substitute  d*tan(14) for "x'

So we have.....

tan(18) = (40- d*tan(14)) /d     ...multiply both sides by d

d*tan(18) = 40 - d*tan(14)       .....add d*tan(14) to both sides

d*tan(18) + d*tan(14) = 40   ....factor the "d" out on the left side

d[tan(18) + tan(14)] = 40         ..........divide both sides by [tan(18) + tan(14)]

d = 40 / [tan(18) + tan(14)] = 69.7 ft

This way....we don't have to worry with "x" at all !!!

Please Sir Headless CPhill.  Can we have 2 pictures.  One for this problem and one for you avatar!

Of my avatar or of the lady looking at the flagple??

Note........they could be one and the same.......

One for each my good Sir!

actully i am not understanding this question , the method of the answer or i can just say that the whole thing going on in here but i would like to give CPhills both anwers thumbs up because he has answered them very nicely as i can see ! so CPhill thumbs up from me for ur both answers ! 

Have you done trigonometry at school Rosala?

Are you saying that you do not understand the question or CPhill's answer?  

If you have done trigonometry at school, and you want to understand this problem then Chris or i can help you understand.  If you haven't done any trig then it will be too difficult for you.

This is as well as I could do using Geogebra....point B is where the lady stands (0, 17.4)...sorry the app "rounded" 17.37 "up"....use your imagination......angle CBA = 14 and angle ABC = 18...the distance the woman is from the flagpole is shown by segment "a" = 69.7........

Rosala...i've provided a somewhat "lousy" picture that demonstrates the situation.....after looking at it...I'm even confused myself !!!!!   (not really....)

Thanks for the "Thumbs-Up,"   anyway........

no i havent done this topic at all in school melody and thanks for the pic CPhill but all of this is just going above my head ! 

no i havent done this topic at all in school melody and thanks for the pic CPhill but all of this is just going above my head ! 

That's the way most flagpoles are......above our heads.......

yes u r right !

THank you so much Cphil that made alot of sense. you saved me from a nervous break down!

Thank you Chris.

I have given you athumbs up on you diagram too.  

this pic is nice melody 1 thumbs up for these cute little creatures ! 

Thank you Rosala,

In the picture there is a Kangaroo, a Kookaburra and a Frilled Neck Lizard.  They are all native to Australia.

This is what a kookaburra sounds like

https://www.youtube.com/watch?v=Qix6oUxim3Q

ur welcome melody !