March 19th, 2017, 05:22 PM  #1 
Newbie Joined: Feb 2017 From: East U.S. Posts: 23 Thanks: 0  Related Rates
A patrol car is parked 30 feet from a long warehouse (see figure). The revolving light on top of the car turns at a rate of 30 revolutions per minute. How fast is the light beam moving along the wall when the beam makes angles of θ = 45°, θ = 60°, and θ = 80° with the line perpendicular from the light to the wall? (Round your answers to two decimal places.) Things I know: 1) tan(θ) = x/30 OR 1/30(x) 2) d(θ)/dt = 60pi 3) I need dx/dt I also know there's something to do with sec^2(θ) and/or cos((45/60/80))^2 I just wanna learn this for myself, it's not for a class. It's super frustrating. Please help... Thanks! 
March 19th, 2017, 05:35 PM  #2 
Math Team Joined: Jul 2011 From: Texas Posts: 2,298 Thanks: 1129 
$30 \, rev/min \, = 60\pi \, rad/min$ $\tan{\theta} = \dfrac{x}{30}$ $\sec^2{\theta} \cdot \dfrac{d\theta}{dt} = \dfrac{1}{30} \cdot \dfrac{dx}{dt}$ $\dfrac{30}{\cos^2{\theta}} \cdot \dfrac{d\theta}{dt} = \dfrac{dx}{dt}$ sub in $60\pi \, rad/min$ for $\dfrac{d\theta}{dt}$ and the desired value(s) of $\theta$ into $\cos^2{\theta}$ to calculate $\dfrac{dx}{dt}$. note $\dfrac{dx}{dt}$ will be in units of ft/min 
March 23rd, 2017, 11:56 AM  #3  
Newbie Joined: Feb 2017 From: East U.S. Posts: 23 Thanks: 0  Quote:
 
March 25th, 2017, 05:15 AM  #4 
Math Team Joined: Jul 2011 From: Texas Posts: 2,298 Thanks: 1129  
March 25th, 2017, 06:00 PM  #5 
Newbie Joined: Feb 2017 From: East U.S. Posts: 23 Thanks: 0  Never mind that. So I plugged in everything you said and got "20491.38" but that's not the right answer according to this website I'm using. This is exactly what I put into my calculator 30/cos(45)^2 which gives me "108.71015", then I multiply all of that by 60pi to get "20491.38" (rounded to two decimal places.) What am I doing wrong? 
March 25th, 2017, 07:28 PM  #6 
Math Team Joined: Jul 2011 From: Texas Posts: 2,298 Thanks: 1129 
If you input degrees, you calculator should be in degree mode ... $\dfrac{30}{\cos^2(45^\circ)} = 60$ 

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