Rate of Change when the boat is 250 ft from the cliff

[mikeskier]

Ok i need to do this problem for calculus and cant figure it out...
its a rate of change problem..

A man is standing on top of a cliff 250 feet up. he is watching a boat coming towards the cliff at 20 ft/sec. what is the rate of chage when the boat is 250 feet from the cliff.

I did so far

Tan[theta]= X[divide]250

sec[squared][theta] [times][theta] prime = 1[divide]250 [times]X prime

Sec[squared][theta][times][theta]Prime = 1[divide]250 [times] 20


and i know you have to do pathag. therom but have:

250[squared] + 250[squared] = 125000 and im stuck from there????


Please Help!

 

[arthur ohlsten]

I find it useful to sketch the problem.
On a xy grid make a mark at x=0 y=250
make a second mark at x=250 y=0
define the angle theta,@, to be the angle subtended at the mark x=0 y=250.

tan@= x/250
sim@/cos@= x/250
take the derivative
cos@[cos@]+sin@sin@] / cos^2@ = 1/250 dx/dt
250/cos^2@ d@/dt=dx/dt
but dx/dt = 20 ft/sex
250 ft/cos^2@ d@/dt=20 ft/sec
but at the time of interest @=45 degrees
250 ft [2] d@/dt =20 ft/sec
d@/dt = 1/25 radians/sec

Arthur