Please help me with this calc im so confused?

Question

A Particle moves on the ellipse 9x^2+16y^2=144 such that dx/dt =4y. FInd dy/dt. At what point is dy/dt a maximum

what is x? Because you need x to substitute in 18xdx/dt+32ydy/dt

Answer 1

Take the derivitive of your equation with respect to t
d(9x^2+16y^2)/dt = 18x dx/dt + 32y dy/dt = 0

They give you dx/dt = 4y.
Substitute this in the above equation and then solve for dy/dt.

The maximum dy/dt will be pretty easy to find from the equation you get.

Answer 2

I find that 18x dx + 16 y dy = 0
question says dx = 4 y dt

so 18x 4 y dt + 16 y dy = 0
so (9/2) x dt + dy = 0
so dy/dt = -(9/2) x

by a funny coincidence dy/dt is a constant times x,
just as dx/dt is a constant times y.
that "coincidence" would be just a property of the
ellipse equation

dy/dt is a maximum when x is a minimum

for the ellipse as stated,
minimum value of x is -4
(if x < -4, y^2 has to be <0 to satisfy the ellipse equation,
and I guess here we're only looking at "real number" solutions)


so maximum value for dy/dt is 18