A particle moves along the curve `6y=x^3+2`. Find the points on the curve at which the y-coordinate is changing 8 times as fast as the x-coordinate.

Explanation

Given`(dy)/(dt) = 8 (dx)/(dt)` …(1)
Equation of curve `6y = x^(3) +2`
`rArr 6 (dy)/(dt) = 3x^(2) (dx)/(dt)`
`rArr 6(8(dx)/(dt)) = 3x^(2) (dx)/(dt)`
` rArr x^(2) = 16`
` rArr x= pm 4`
at x = 4 ` y = (4^(3)+2)/6 =11`
at x = - 4 ` y ((-4)^(3) + 2)/6 (-31)/3`
`:." Required points " = (4, 11) and (-4, - (31)/3)`