Question

I need help with this Calculus problem

Answer 1

Hmm are you allowed to make multiple guesses?
Or just one try each? :U

Answer 2

I came up with something and want to burn a guess and see if it's correct before I explain it lol.

Answer 3

I have 3 more tries

Answer 4

Ok then I guess let's start with what we know.

\[\Large\bf\sf \cot \theta\quad=\quad \frac{x}{5}\]Taking the derivative of this expression (with respect to time) will give us an idea of what's going on. It will make a d(theta)/dt show up, which is what we want.
Do you remember your cotangent derivative?

Answer 5

\[\frac{ -1 }{ (\sin(x)^2) }\]

Answer 6

\[ \frac{d}{dx} cot^{-1}(x) = \frac{-1}{1 + x^2}\]

Answer 7

so I would think by chain rule you should have:

\[\frac{1}{5}*\frac{-1}{1+(\frac{x}{5})^2}\]

Answer 8

\[\Large\bf\sf -\csc^2\theta\;\frac{d \theta}{dt}\quad=\quad \frac{1}{5}\cdot\frac{dx}{dt}\]Yes, 1/sin^2, good.

Answer 9

Chain rule is giving us derivative terms.
Since our derivative was with respect to t, we get both a derivative term for theta AND x.

Answer 10

I'm assuming the speed of the plane as something to do with x...
See how the plane is moving in the x direction?
x get's smaller as the plane moves left.

So I would think of the velocity as dx/dt = -365

Answer 11

Hopefully I'm interpreting that correctly.

Answer 12

Solve for d(theta)/dt.
Use your triangle to plug in a value for your trig function part.
(Find the hypotenuse using Pythagorean Theorem).

Answer 13

I'm still lost...
Don't I just have to plug in the x for an 8 for the derivative of arccot(x/5)?

Answer 14

I got:

\[\frac{d\theta}{dt} = -\frac{5}{x^2 + 25}\frac{dx}{dt} \]
\[\frac{d\theta}{dt} = -\frac{5}{x^2 + 25}(-365) \]
\[\frac{d\theta}{dt} = \frac{1825}{x^2 + 25} \]

Answer 15

not 100% on it though, but you can give it a try.

Answer 16

Nope, didn't work. Nice try though

Answer 17

damn... let me see if I can figure out what @zepdrix was doing

Answer 18

That didn't work? Hmm that's the same thing I came up with :[

Answer 19

I got 20.506 for x = 8

Answer 20

when you simplify what zepdrix did, you get:

\[ \frac{d\theta}{dt} = 73sin^2\theta\]

now plugin you're equation for theta in terms of x maybe?

Answer 21

20.506 worked?

Answer 22

no, 20.506 didn't work

Answer 23

so you mean, 73sin(8)^2?

Answer 24

no, i meant:
\[ \frac{d\theta}{dt} = 73*sin^2(cot^{-1}(x/5))\]

but guess what, that makes:
\[\frac{d\theta}{dt} = \frac{1825}{x^2 + 25}\]

same as before.

Answer 25

The fact that we were not TOLD what units x is measured in is really annoying.
I'm assuming it's in miles, but maybe they meant x=5 kilometers and are playing a trick on us or something :(

Answer 26

maybe its the negative for velocity?

Answer 27

try switching the sign, its all I got left :P:
\[\frac{d\theta}{dt} = -\frac{1825}{x^2 + 25}\]

Answer 28

Here is a different version but with the correct answer if it helps

Answer 29

oo nice something to compare :)

Answer 30

that shows the equation being correct:

\[\frac{d\theta}{dt} = \frac{5}{x^2 + 25} * 394\]
\[\frac{d\theta}{dt} = \frac{1970}{x^2 + 25}\]
\[\frac{d\theta}{dt} = \frac{1970}{8^2 + 25}\]
\[\frac{d\theta}{dt} = \frac{1970}{64 + 25}\]
\[\frac{d\theta}{dt} = \frac{1970}{89}\]
\[\frac{d\theta}{dt} = 22.134831\]

Answer 31

Ya that's really strange.
We did the problem correctly...

Answer 32

Oh your problem says \(\Large\bf\sf 356\)......

we've been using \(\Large\bf\sf 365\)................. Could that have been the issue? :(
Oh boy...

Answer 33

That one's on me t.t ugh..

Answer 34

ah crap.

Answer 35

Did I input it wrong then?

Answer 36

Yes, it's my fault though.
I posted in my earlier comment, that the velocity should be dx/dt = -365.

When in fact your problem page reads 356.

Answer 37

\[\Large\bf\sf \frac{d \theta}{dt}\quad=\quad -\frac{5}{x^2+25}\cdot (-356)\]

Answer 38

x=8 is giving me exactly 20.

Answer 39

It worked! Thank you guys so much. So who should I give the medal to?
The answers were 20 and 35.6

Answer 40

Double check my work though :U
That is if you still have some guesses available.

Answer 41

We share XD