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March 19th, 2017, 05:22 PM   #1
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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!
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March 19th, 2017, 05:35 PM   #2
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$30 \, rev/min \, = 60\pi \, rad/min$

$\tan{\theta} = \dfrac{x}{30}$

$\sec^2{\theta} \cdot \dfrac{d\theta}{dt} = \dfrac{1}{30} \cdot \dfrac{dx}{dt}$

$\dfrac{30}{\cos^2{\theta}} \cdot \dfrac{d\theta}{dt} = \dfrac{dx}{dt}$

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
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March 23rd, 2017, 11:56 AM   #3
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Originally Posted by skeeter View Post
$30 \, rev/min \, = 60\pi \, rad/min$

$\tan{\theta} = \dfrac{x}{30}$

$\sec^2{\theta} \cdot \dfrac{d\theta}{dt} = \dfrac{1}{30} \cdot \dfrac{dx}{dt}$

$\dfrac{30}{\cos^2{\theta}} \cdot \dfrac{d\theta}{dt} = \dfrac{dx}{dt}$

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
Thank you, but how do I get cos^2(θ) again?
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March 25th, 2017, 05:15 AM   #4
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Thank you, but how do I get cos^2(θ) again?
what do you mean?
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March 25th, 2017, 06:00 PM   #5
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what do you mean?
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?
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March 25th, 2017, 07:28 PM   #6
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If you input degrees, you calculator should be in degree mode ...

$\dfrac{30}{\cos^2(45^\circ)} = 60$
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