October 10th, 2016, 01:11 PM  #1 
Member Joined: Mar 2015 From: uk Posts: 33 Thanks: 1  Height of a water tower problem
This question is at the end of a section on limits and differentiation, A landmark on a distant hill is 'x' metres from a water tower. The angle of elevation from the top of the tower is observed to be '0' (0 is theta) degrees whereas the angle from the foot of the tower is observed to be 0+h degrees a)how high is the water tower b)show that if 'h' is small then the height of the water tower is approx (Pi)xh / 180(cosn)^2 I've managed to do the first part. If the height of the landmark is 't' and the height of the water tower is 'n' then, tan (0+h) = t/x and tan 0 = (t  n) / x Eliminating t, t = xtan(0+h) = xtan0 +n So the height of the water tower 'n' is x(tan(0+h)  tan0) Can anyone show me how to do the second part. I don't understand how (Pi) and 180 can become involved?? 
October 11th, 2016, 01:06 PM  #2 
Global Moderator Joined: May 2007 Posts: 6,732 Thanks: 689 
Statement if b) doesn't look right. How can a dumber with dimension of length appear as an argument for cosn? Pi and 180 become involved, since trig functions use radians as a "natural" argument  for example sin(x) ~ x when x is in radians. 
October 11th, 2016, 01:53 PM  #3 
Senior Member Joined: Sep 2015 From: USA Posts: 2,408 Thanks: 1310 
if $h$ is small then $\tan(\theta+h) \approx \tan(\theta)+\left. \dfrac{d}{d\theta}\tan(\theta)\right_{\theta=h}$ $\tan(x+h) \approx \tan(\theta)+\sec^2(h)$ the rest follows 
October 12th, 2016, 01:45 PM  #4 
Global Moderator Joined: May 2007 Posts: 6,732 Thanks: 689  This assumes h is in radians, not in degrees. h in degrees is where $\displaystyle \frac{\pi}{180}$ comes into play.

October 20th, 2016, 02:29 AM  #5 
Member Joined: Mar 2015 From: uk Posts: 33 Thanks: 1 
Thanks for the replies. Having looked at it some more I think this is how it is done n=x(tan(0+h)  tan0) now dy/dx is defined as limit as h tends to zero of [f(x+h)  f(x)] / h therefore n=xh [lim (h tends to zero) (tan(0+h)  tan0 / h)] so n=xh (d/d0 tan0) = xh / (cos0)^2 degrees or (pi)xh / 180(cos0)^2 degrees 

Tags 
height, problem, tower, water 
Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
When a drop of water falls into water, where do the splashes come from?  Ganesh Ujwal  Physics  2  January 3rd, 2015 06:47 AM 
Water Tank Problem  mr. jenkins  Calculus  1  May 26th, 2014 01:09 AM 
Height of the tower  rakmo  Algebra  3  March 28th, 2013 05:20 AM 
A word problem about salty water (using limits)  thearae  Calculus  7  September 26th, 2012 03:51 AM 
Ordnance Calculation,height of a tower  manich44  Algebra  9  June 23rd, 2009 09:23 AM 