How fast is the distance from the particle to the origin changing at this instant?

Question

A particle is moving along the curve y=sqrt(x) As the particle passes through the point (4,2), its x-coordinate increases at a rate of 3cm/s. How fast is the distance from the particle to the origin changing at this instant?

___ cm/s

Answer 1

Square of distance d² = x² + y² = x² + x; (d²)' = 2dd' = 2xx' + x'; d' = (2xx' + x')/(2d). At point (4,2) x = 4,

d = √4² + 2² = √20 = 2√5 cm; d' = (2·4·3 + 3)/(2√5) = 27/(2√5) = 27√5/10 cm/s = 6.0374 cm/s