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October 10th, 2016, 01:11 PM   #1
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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??
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October 11th, 2016, 01:06 PM   #2
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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.
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October 11th, 2016, 01:53 PM   #3
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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
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October 12th, 2016, 01:45 PM   #4
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Quote:
Originally Posted by romsek View Post
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
This assumes h is in radians, not in degrees. h in degrees is where $\displaystyle \frac{\pi}{180}$ comes into play.
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October 20th, 2016, 02:29 AM   #5
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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
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