Question

assume x=x(t) and y=y(t). Find dx/dt if x^2(y-6) = 12y + 3 and dy/dt = 2 when x = 5 and y = 12

Answer 1

I'm pretty lost in general about this problem but I think it's supposed to use differential equations!

Answer 2

Simply differentiate it to get
\[\Large 2x \frac{dx}{dt}(y-6) + x^2 (\frac{dy}{dt})=12 \frac{dy}{dt}\]
solve for dx/dt by rearranging terms
x=5,y=12
dy/dt=2

Answer 3

So when you say solve for dx/dt by rearranging, would it be:

\[(12-6)+5 ^{2}(2)-12(2)=2(5)\frac{ dx }{ dt } ? \]

Or is that incorrect? I feel 100% sure that I made an error somewhere but I'm not sure how to separate it honestly.

Answer 4

\[\Large \frac{dx}{dt}=\frac{12 \frac{dy}{dt}-x^2 \frac{dy}{dt}}{2x}\]

Answer 5

now put all the values

Answer 6

\[\frac{ dx }{ dt }= \frac{ 24 - 50 }{ 10 } = \frac{ -26 }{ 10 }= \frac{ -13 }{ 5 }\]

Is that correct?

Answer 7

or should it be 24+50

Answer 8

-13/5 is absolutely correct mate :) well done

Answer 9

Here's the confusing part though. The only viable answer choices on my test (I've already taken it btw) are:

\[-\frac{ 20 }{ 13 } | -\frac{ 13 }{ 30 } | \frac{ 13 }{ 20 } | \frac{ 20 }{ 13 }\]

I circled the third option 13/20 and got the question wrong and -13/5 doesn't match any solution unfortunately 8(

Answer 10

Anyone have an idea where I went wrong?

Answer 11

or what went wrong?

Answer 12

x^2(y-6) = 12y + 3

2x(y-6) + x^y' = 12y'

y' = (2x(y-6))/(12-y^2)

y' at (5, 12) = -60/13

Answer 13

dy/dx = dy/dt / dx/dt

-60/13 = 2/A

solve A

Answer 14

-13/30! I understand it now! Thanks !

Answer 15

np :) im still wondering why the other method was not giving the same answer :/