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 March 19th, 2017, 05:22 PM #1 Member   Joined: Feb 2017 From: East U.S. Posts: 40 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: 2,717 Thanks: 1372 $30 \, rev/min \, = 60\pi \, rad/min$ $\tan{\theta} = \dfrac{x}{30}$ $\sec^2{\theta} \cdot \dfrac{d\theta}{dt} = \dfrac{1}{30} \cdot \dfrac{dx}{dt}$ $\dfrac{30}{\cos^2{\theta}} \cdot \dfrac{d\theta}{dt} = \dfrac{dx}{dt}$ 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 Thanks from Maschke, SenatorArmstrong and nbg273
March 23rd, 2017, 11:56 AM   #3
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 Originally Posted by skeeter $30 \, rev/min \, = 60\pi \, rad/min$ $\tan{\theta} = \dfrac{x}{30}$ $\sec^2{\theta} \cdot \dfrac{d\theta}{dt} = \dfrac{1}{30} \cdot \dfrac{dx}{dt}$ $\dfrac{30}{\cos^2{\theta}} \cdot \dfrac{d\theta}{dt} = \dfrac{dx}{dt}$ 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?

March 25th, 2017, 05:15 AM   #4
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 Originally Posted by nbg273 Thank you, but how do I get cos^2(θ) again?
what do you mean?

March 25th, 2017, 06:00 PM   #5
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 Originally Posted by skeeter 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?

 March 25th, 2017, 07:28 PM #6 Math Team   Joined: Jul 2011 From: Texas Posts: 2,717 Thanks: 1372 If you input degrees, you calculator should be in degree mode ... $\dfrac{30}{\cos^2(45^\circ)} = 60$
March 25th, 2017, 10:32 PM   #7
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 Originally Posted by skeeter If you input degrees, you calculator should be in degree mode ... $\dfrac{30}{\cos^2(45^\circ)} = 60$
Ok, so in that case, I got "11309.73(rounded)" as my answer and it's still not working..... Could you please just tell me what the answer should be? I'm sorry to annoy you on the forums, but I've been trying to figure this out for 3 whole days and it's driving me crazy.....

 March 25th, 2017, 10:53 PM #8 Senior Member   Joined: Feb 2016 From: Australia Posts: 1,533 Thanks: 513 Math Focus: Yet to find out. It's likely your calculator isn't set to degrees, as skeeter said. What is the model of your calculator?
March 26th, 2017, 12:33 AM   #9
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 Originally Posted by Joppy It's likely your calculator isn't set to degrees, as skeeter said. What is the model of your calculator?
TI-84 PLUS, but I set it to degree mode. "30/cos(45)^2" = 60. Then I multiplied 60 times 60pi, but that's not the right answer.

I'm trying to find dx/dt using this formula:

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

So I plug everything in and it's not right... It should be 60*60pi, right?

March 26th, 2017, 12:42 AM   #10
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 Originally Posted by nbg273 TI-84 PLUS, but I set it to degree mode. "30/cos(45)^2" = 60. Then I multiplied 60 times 60pi, but that's not the right answer. I'm trying to find dx/dt using this formula: $\dfrac{30}{\cos^2{\theta}} \cdot \dfrac{d\theta}{dt} = \dfrac{dx}{dt}$ So I plug everything in and it's not right... It should be 60*60pi, right?
$\dfrac{dx}{dt} = \dfrac{30}{\cos^2{(\frac{\pi}{4}})} \dfrac{feet}{\cancel{radian}} \cdot \dfrac{60 \pi \ \cancel{radian}}{minute}$

Note that $45^{\circ} \cdot \dfrac{2 \pi}{360^{\circ}} = \dfrac{\pi}{4} \ radians$.

Last edited by Joppy; March 26th, 2017 at 12:48 AM. Reason: lost pi

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