writing rate of change equations

[poppy]

Hi
Im trying to write an equation for the question below. i did write one but am pretty sure it is wrong because I need to differentiate arctan in it and we have not been taught that yet. Could someone please point me in the right direction with writing it?

A telescope is 75m above water level on a cliff and a boat is approaching at 6m/s. what is the rate of change of angle of the telescope when the boat is 75m from shore.

I thought it might be y=arctan(75/x) because the angle specified is found using arctan(opposite/addjacent).

Any suggestions appreciated

 

[galactus]

You do not have to use arctan, but you can if you must.

\(\displaystyle tan{\theta}=\frac{75}{x}\)

Differentiate both sides w.r.t time:

\(\displaystyle sec^{2}{\theta}\frac{d\theta}{dt}=\frac{-75}{x^{2}}\frac{dx}{dt}\)

When the boat is 75 m from the cliff, then the angle is \(\displaystyle \frac{\pi}{4}\)

Plug in your knowns and solve for \(\displaystyle \frac{d\theta}{dt}\)