# Thread: Right Triangle Trig word problem

1. ## Right Triangle Trig word problem

I'm stuck on this...sorry, there's a diagram I dont' have time to scan, but hopefully it is clear enough:

"A flagpole is located at the edge of a sheer 50ft cliff at the bank of a river of width 40 feet. An observer on the opposite side of the river measures an angle of 9 degrees between her line of sight to the top of the flagpole and her line of sight to the top of the cliff. Find the height of the flagpole."

So we know that the cliff is 50feet, but we don't know what the flagpole is so that would be 50+x correct?
And there's a right-angle with the base of 40 feet and height of 50 feet, so the hypotenuse would be 60 feet I believe.
But I don't think that helps me. Very confused...

2. Originally Posted by tiar
I'm stuck on this...sorry, there's a diagram I dont' have time to scan, but hopefully it is clear enough:

"A flagpole is located at the edge of a sheer 50ft cliff at the bank of a river of width 40 feet. An observer on the opposite side of the river measures an angle of 9 degrees between her line of sight to the top of the flagpole and her line of sight to the top of the cliff. Find the height of the flagpole."

So we know that the cliff is 50feet, but we don't know what the flagpole is so that would be 50+x correct?
And there's a right-angle with the base of 40 feet and height of 50 feet, so the hypotenuse would be 60 feet I believe.
But I don't think that helps me. Very confused...
You don't actually need to find the length of the hypotenuse. If you assume the observer has no height, then the angle between her line of sight to the base of the cliff to the top of the cliff is arctan(50/40) degrees, and the angle between her line of sight to the base of the cliff to the top of the flagpole is [arctan(50/40) + 9] degrees. Can you figure it out after calculating that angle?

3. Hello, tiar!

A flagpole is located at the edge of a sheer 50ft cliff at the bank of a river 40 feet wide.
An observer on the opposite side of the river measures an angle of 9 degrees between
her line of sight to the top of the flagpole and her line of sight to the top of the cliff.
Find the height of the flagpole.
Code:
                          o A
* |
*   |
*     | h
*       |
*         o B
*       *   |
* 9°  *       |
*   *           | 50
* *  θ            |
P o - - - - - - - - - o C
40

The flagpole is $\displaystyle h = AB.$
The cliff is $\displaystyle BC = 50.$
The observer is at $\displaystyle P\!:\;\;PC = 40.$
$\displaystyle \angle APB = 9^o.$
Let $\displaystyle \theta = \angle BPC.$

In right triangle $\displaystyle BCP\!:\;\tan\theta = \tfrac{50}{40} = \tfrac{5}{4}$

In right triangle $\displaystyle ACP\!:\;\;\tan(\theta + 9^o) \:=\:\frac{h+50}{40}\;\;{\color{blue}[1]}$

. . Note that: .$\displaystyle \tan(\theta + 9^o) \:=\:\frac{\tan\theta + \tan9^o}{1-\tan\theta\tan9^o} \;=\;\frac{\frac{5}{4} + \tan9^o}{1 - \frac{5}{4}\tan9^o}$ .$\displaystyle = \:\frac{5+4\tan9^o}{4-5\tan9^o}$

Substitute into [1]: .$\displaystyle \frac{5+\tan9^o}{4-5\tan9^o} \;=\;\frac{h+50}{40} \quad\Rightarrow\quad h \;=\;40\cdot\frac{5 + 4\tan9^o}{4-5\tan9^o} - 50$

Therefore: .$\displaystyle h \;=\;20.24190952 \;\approx\;20.24\text{ ft}$