# 3d word problem

1. Apr 4, 2015

### Appleton

1. The problem statement, all variables and given/known data
A vertical mast stands on the north bank of a river with straight parallel banks running from east to west. The angle of elevation of the top of the mast is α when measured from a point A on the south bank distant 3a to the east of the mast and β when measured from another point B on the south bank distant 5a to the west mast. prove that the height of the mast is
$4a/(cot^2\beta-cot^2\alpha)^\frac{1}{2}$

2. Relevant equations

3. The attempt at a solution
Let the height of the mast be h.
Let C be the base of the mast.
Let D be the point at which the perpendicular from BA to C divides BA.

$BC = \frac{h}{tan β}\\\\ CA = \frac{h}{tan α}\\\\$
By Pythagoras' theorem

$CD = \sqrt{(\frac{h}{tan β})^2 - (5a)^2)}\\\\ CD = \sqrt{(\frac{h}{tan α})^2 - (3a)^2)}\\\\$
So

$\sqrt{(\frac{h}{tan β})^2 - (5a)^2)} = \sqrt{(\frac{h}{tan α})^2 - (3a)^2)}\\\\\ (\frac{h}{tan β})^2 - (\frac{h}{tan α})^2 = (5a)^2 - (3a)^2\\\\ \frac{h^2 tan^2 α - h^2 tan^2 β }{(tan^2 β) (tan^2 α)} = 16a^2\\ h = 4a\sqrt{\frac{(tan^2 β) (tan^2 α)}{tan^2 α - tan^2 β }}$

At this point I figure that either the question is floored or i've made a mistake. Usually it's the latter.

Last edited: Apr 4, 2015
2. Apr 4, 2015

### SammyS

Staff Emeritus
... or you have to take this a bit further.

What is $\displaystyle\ \left(\frac{(tan^2 β) (tan^2 α)}{tan^2 α - tan^2 β }\right)^{-1}\$ ?

3. Apr 4, 2015

### Appleton

Ah yes, i'm kicking myself. Thanks for the nudge.

Last edited: Apr 4, 2015