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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?
 one year ago
 one year 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?
 one year ago
 one year ago

This Question is Closed

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

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

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

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

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

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

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

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

NickR Group TitleBest ResponseYou've already chosen the best response.0
1/x^2*dx/dt
 one year ago

Algebraic! Group TitleBest ResponseYou've already chosen the best response.1
great:)
 one year ago

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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