A particle moves counterclockwise around the ellipse 3x2 + 8y2 = 11. At what rate is the x-coordinate changing when the particle passes the point (1, 1) if its y-coordinate is increasing at a rate of 8 ft/s?

D(x) /dt  =  - 21,33 ft/s

Step-by-step explanation:

We have an equation, in which both  x  and  y coodinates are function of t, then we take derivatives on both sides of the equation to obtain

3*x²   +  8*y²  =  11

6* x * D(x) /dt  +  16* y * D(y)/dt  = 0

6* x * D(x) /dt   =  -  16* y * D(y)/dt       (1)

Now from problem statement we know:

D(y) /dt  = 8 ft/s

And we are loking for D(x)/dt  = ?? when particle passes the point ( 1,1)

x = 1      y  =  1

Plugging these values in equaton (1)

6* x * D(x) /dt   =  -  16* y * D(y)/dt  

6* D(x) /dt  = - 16* 8

D(x) /dt  =  - 128 /6      ⇒    D(x) /dt  =  - 21,33 ft/s