Question

A particle moves along the curve . Find the points on the curve at which the y -coordinate is changing 8 times as fast as the x -coordinate.

Answer 1

It is given that a particle moves along the curve,

6y= x 3 +2(1)

The y-coordinate is changing 8 times as fast as x-coordinate. Therefore,

dy dt =8 dx dt (2)

Differentiate both sides of equation (1) with respect to t,

6 dy dt =3 x 2 dx dt

Substitute dy dt =8 dx dt from equation (2) in the above equation.

6( 8 dx dt )=3 x 2 dx dt 3 x 2 =48 x 2 =16 x=±4

Substitute x=4 in equation (1),

6y= 4 3 +2 6y=64+2 6y=66 y=11

Now, substitute x=−4 in equation (1),

6y=− 4 3 +2 6y=−62 y= −62 6 = −31 3

Thus, the required points on the curve are ( 4,11 ) and ( −4, −31 3 ).