anonymous
  • anonymous
A kite 100 ft above the ground moves horizontally at a speed of 8 ft/s. At what rate is the angle (in radians) between the string and the horizontal decreasing when 200 ft of string have been let out? i got -8/400 but it was wrong
Mathematics
  • Stacey Warren - Expert brainly.com
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katieb
  • katieb
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freckles
  • freckles
|dw:1434325170769:dw| picture looks something like this
freckles
  • freckles
|dw:1434325227788:dw|
freckles
  • freckles
"A kite 100 ft above the ground moves horizontally at a speed of 8 ft/s. At what rate is the angle (in radians) between the string and the horizontal decreasing when 200 ft of string have been let out? i got -8/400 but it was wrong " --- Moves horizontally at a speed of 8ft/s this means x'=8ft/s we are looking for theta (in rad) when y=200ft

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freckles
  • freckles
well what is theta when y=200ft
freckles
  • freckles
what trig function can you use here
freckles
  • freckles
|dw:1434325371019:dw|
freckles
  • freckles
hint you have opp and hyp measurements given
anonymous
  • anonymous
ok i did sin of 100/200 and got .479
freckles
  • freckles
arcsin(1/2)? which is pi/6
anonymous
  • anonymous
why arcsin?
freckles
  • freckles
to solve sin(theta)=1/2 you normally take arcsin( ) of both sides since arcsin(sin(x))=x where x is between -pi/2 and pi/2 |dw:1434325633246:dw| anyways we are going to use x (because we are given x's rate) and theta for sure and I will go ahead and use that constant side \[\tan(\theta)=\frac{100}{x}\] differentiate both side
anonymous
  • anonymous
sec^2x=0?
freckles
  • freckles
how did you get that equation?
freckles
  • freckles
sec is never 0
anonymous
  • anonymous
im confused...
freckles
  • freckles
Like can you tell me what is confusing you please? Because I don't know how to help if I don't know where to start to help.
anonymous
  • anonymous
the derivative of tan=100/x thats where i got confused
freckles
  • freckles
well the equation is actually \[\tan(\theta)=\frac{100}{x} \text{ or you can write it as } \\ \tan(\theta)=100x^{-1}\] so you don't have to use quotient rule if you prefer
freckles
  • freckles
you will need to use chain rule on both sides though
freckles
  • freckles
do you know how to evluate either one of these: \[\frac{d \tan(t)}{dt}=? \\ \frac{d(t^{-1})}{dt}=?\]
freckles
  • freckles
for the first it is easy just to remember derivative of tan(t) w.r.t t is sec^2(t) and the second one I asked you about can be done by power rule have you every used power rule before? I hope so because this question is about application of chain rule
freckles
  • freckles
\[\tan(\theta)=100x^{-1} \\ \text{ differentiate both sides } \\ \sec^2(\theta) \cdot \frac{d \theta}{d t}=100 \cdot \frac{d(x^{-1})}{dt}\] you still to differentiate the right hand side
anonymous
  • anonymous
would it be 100/-x^2?
freckles
  • freckles
don't forget you need to write times dx/dt
freckles
  • freckles
\[\sec^2(\theta) \cdot \frac{d \theta}{ d t}=-\frac{100}{x^2} \cdot \frac{dx }{dt}\]
freckles
  • freckles
plug in all the values you found and are given and solve for dtheta/dt
anonymous
  • anonymous
would it be sec^2(pi/6)*dtheta/dt=- 100/1ooroot3* 200?
freckles
  • freckles
well the x has a square on it and I thought dx/dt was given as 8ft/sec why you put 200 for dx/dt? dx/dt represents the rate and which the kite moves horizontally
freckles
  • freckles
\[\sec^2(\frac{\pi}{6}) \cdot \frac{ d \theta}{ d t}=\frac{-100}{(100 \sqrt{3})^2} \cdot 8\]
anonymous
  • anonymous
i solved the right hand side but then how do i divide by the sec^2 im havinf trouble calculating that?
freckles
  • freckles
1/sec(k)=cos(k)
freckles
  • freckles
\[\frac{d \theta}{dt} =\frac{1}{\sec^2(\frac{\pi}{6})} \frac{-100}{(100 \sqrt{3})^2} \cdot 8 \\ \frac{d \theta}{ dt}=\cos^2(\frac{\pi}{6}) \frac{-100}{(100\sqrt{3})^2} \cdot 8\]
anonymous
  • anonymous
okay i got -.o2?
freckles
  • freckles
ok and i have -1/50 those should be equal
freckles
  • freckles
don't forget to put your units
freckles
  • freckles
have fun
anonymous
  • anonymous
wait
anonymous
  • anonymous
my answer was -8/400 and i got it wrong?
anonymous
  • anonymous
or is it not supposed to be negative?
freckles
  • freckles
you can try without did you put the units though ?
anonymous
  • anonymous
the units are already tehre rad/s so i just had to type my answer in
freckles
  • freckles
k
freckles
  • freckles
have you got it yet?
anonymous
  • anonymous
i only have one more try so im scared to type it and get it wrong
freckles
  • freckles
ok don't know what to tell you we have the answer above says it is decreasing at a rate of 1/50 rad/sec
anonymous
  • anonymous
i know this thing is weird -_- but thank you for your help though!!
anonymous
  • anonymous
it was supposed to be positive got it right!!!

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