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NickR
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A kite 50ft above the ground moves horizontally at a speed of 8ft/s. At what rate is the angle between the string and the horizontal decreasing when 300ft of string has been let out?
 2 years ago
 2 years ago
NickR Group Title
A kite 50ft above the ground moves horizontally at a speed of 8ft/s. At what rate is the angle between the string and the horizontal decreasing when 300ft of string has been let out?
 2 years ago
 2 years ago

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Algebraic! Group TitleBest ResponseYou've already chosen the best response.1
dw:1351143411523:dw
 2 years ago

Algebraic! Group TitleBest ResponseYou've already chosen the best response.1
tan(theta) = 50/x differentiate both sides with respect to time find x when sqrt(x^2+50^2) =300 (*spoiler*) find theta when x= sqrt(300^2 50^2) plug x, theta and dx/dt into the result of the differentiation to find d(theta)/dt
 2 years ago

NickR Group TitleBest ResponseYou've already chosen the best response.0
can you explain the diferrentiatie both sides with respect to time part?
 2 years ago

Algebraic! Group TitleBest ResponseYou've already chosen the best response.1
I guess, it's just implicit differentiation really...or the chain rule... or whatever you are comfortable thinking of it as... d/dt ( tan ( f(t) ) =
 2 years ago

Algebraic! Group TitleBest ResponseYou've already chosen the best response.1
derivative of the 'outside' is...?
 2 years ago

NickR Group TitleBest ResponseYou've already chosen the best response.0
so it's x = 50/sec^2(theta)?
 2 years ago

Algebraic! Group TitleBest ResponseYou've already chosen the best response.1
yep, so sec^2 (f(t) ) * f '(t)
 2 years ago

Algebraic! Group TitleBest ResponseYou've already chosen the best response.1
f(t) is theta yeah, I wrote it that way because theta depends on time and I wanted you to see the chain rule
 2 years ago

Algebraic! Group TitleBest ResponseYou've already chosen the best response.1
so LHS is sec^2(theta) * d(theta) /dt RHS still needs to be differentiated...
 2 years ago

Algebraic! Group TitleBest ResponseYou've already chosen the best response.1
d/dt ( 1/(g(t)) ) =...?
 2 years ago

Algebraic! Group TitleBest ResponseYou've already chosen the best response.1
1/ ..........
 2 years ago

NickR Group TitleBest ResponseYou've already chosen the best response.0
what is g(t) in this case?
 2 years ago

Algebraic! Group TitleBest ResponseYou've already chosen the best response.1
well, just because I used f(t) for theta... now we're talking about x, which is a different function of time...
 2 years ago

Algebraic! Group TitleBest ResponseYou've already chosen the best response.1
so:\[\frac{ d \theta }{dt } = \frac{ 1 }{x^2 \sec^2\theta }\frac{ d x}{ dt }\]
 2 years ago

Algebraic! Group TitleBest ResponseYou've already chosen the best response.1
dx/dt is given.. x and theta are easy to find...
 2 years ago

NickR Group TitleBest ResponseYou've already chosen the best response.0
d(theta)/dt = 8/x^2*sec(theta)
 2 years ago

Algebraic! Group TitleBest ResponseYou've already chosen the best response.1
x= sqrt(300^2 50^2) theta = arctan (50/x)
 2 years ago

NickR Group TitleBest ResponseYou've already chosen the best response.0
wouldn't x = sqrt(300^2 + 50^2)?
 2 years ago

NickR Group TitleBest ResponseYou've already chosen the best response.0
theta = .1629 rads
 2 years ago

Algebraic! Group TitleBest ResponseYou've already chosen the best response.1
so x is longer than the hypotenuse?
 2 years ago

NickR Group TitleBest ResponseYou've already chosen the best response.0
so d(theta)/dt = 8/((295.803)^2*(sec^2(.16744 rads)) which is incorrect.
 2 years ago

Algebraic! Group TitleBest ResponseYou've already chosen the best response.1
8.87E5?
 2 years ago

NickR Group TitleBest ResponseYou've already chosen the best response.0
well you said d(theta)/dt = 1/(x^2*sec^2(theta)) * dx/dt
 2 years ago

Algebraic! Group TitleBest ResponseYou've already chosen the best response.1
you didn't get 8.87E5?
 2 years ago

Algebraic! Group TitleBest ResponseYou've already chosen the best response.1
and it says it's wrong...
 2 years ago

Algebraic! Group TitleBest ResponseYou've already chosen the best response.1
dunno let me look it all over. I don't see any glaring mistakes...
 2 years ago

Algebraic! Group TitleBest ResponseYou've already chosen the best response.1
8.89E5 rad.s/sec .. best I can do... let's see who else is on, might be able to see if I made a mistake...
 2 years ago

RolyPoly Group TitleBest ResponseYou've already chosen the best response.0
May I have the answer to the question?
 2 years ago

Algebraic! Group TitleBest ResponseYou've already chosen the best response.1
go for it.
 2 years ago

NickR Group TitleBest ResponseYou've already chosen the best response.0
@RolyPoly id like the answer too!
 2 years ago

Algebraic! Group TitleBest ResponseYou've already chosen the best response.1
@callisto is checking it over...
 2 years ago

Algebraic! Group TitleBest ResponseYou've already chosen the best response.1
ty @Callisto
 2 years ago

Callisto Group TitleBest ResponseYou've already chosen the best response.1
I... am no good at maths... \[tan \theta = \frac{50}{x}\] Differentiate both sides with respect to x. Probably you won't get: \[\frac{ d \theta }{dt } = \frac{ 1 }{x^2 \sec^2\theta }\frac{ d x}{ dt }\]
 2 years ago

Callisto Group TitleBest ResponseYou've already chosen the best response.1
My bad, I meant with respect to t
 2 years ago

Algebraic! Group TitleBest ResponseYou've already chosen the best response.1
you're saying the differentiation is wrong?
 2 years ago

Callisto Group TitleBest ResponseYou've already chosen the best response.1
\[tan \theta = \frac{50}{x}\] Diff. both sides w.r.t. t \[sec^2\theta \frac{d\theta}{dt} = \frac{50}{x^2} \frac{dx}{dt}\]\[\frac{d\theta}{dt} = \frac{50}{x^2sec^2\theta } \frac{dx}{dt}\]
 2 years ago

Algebraic! Group TitleBest ResponseYou've already chosen the best response.1
forgot the 50 rfl
 2 years ago

Algebraic! Group TitleBest ResponseYou've already chosen the best response.1
you so smart cally
 2 years ago

NickR Group TitleBest ResponseYou've already chosen the best response.0
50*8/((295.803^2)*sec^2(.16744)) = correct answer
 2 years ago

NickR Group TitleBest ResponseYou've already chosen the best response.0
thank you @Callisto and @Algebraic!
 2 years ago

Algebraic! Group TitleBest ResponseYou've already chosen the best response.1
yes thanks @Callisto !
 2 years ago

Callisto Group TitleBest ResponseYou've already chosen the best response.1
\[sec^2\theta=\frac{x^2+50}{x^2}\]You can cancel the x^2 since \[\frac{d\theta}{dt} = \frac{50}{x^2sec^2\theta } \frac{dx}{dt}=\frac{50}{x^2(\frac{x^2+50}{x^2}) } \frac{dx}{dt}\]\[=\frac{50}{(x^2+50) } \frac{dx}{dt}\]And x^2 is easy to find
 2 years ago

NickR Group TitleBest ResponseYou've already chosen the best response.0
only took me 4hrs but i got my question worth 1pt right now to get some sleep
 2 years ago

NickR Group TitleBest ResponseYou've already chosen the best response.0
you guys have a good night thanks again
 2 years ago

Callisto Group TitleBest ResponseYou've already chosen the best response.1
Good night!!~
 2 years ago

robtobey Group TitleBest ResponseYou've already chosen the best response.0
A revised Mathematica solution, posted on 25 October 2012, 22:15 California time, is attached.
 2 years ago
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