Question

A particle is moving along the parabola x^2=4(y+6). As the particle passes through the point (8,10), the rate of change of its y-coordinate is 5 units per second. How fast, in units per second, is the x-coordinate changing at this instant?

Answer 1

You'll need to use differentiation (specifically, implicit differentiation) here.

If x^2 = 4(y+6), differentiating both sides with respect to time t produces the following:

2x (dx/dt) = 4([dy/dt])   (note that (d/dt) 6 = 0)

We need to solve for (dx/dt).  Substitute 8 for x (y does not appear in this latest equation, so we do nothing with y=10).  Substitute the given 5 units/sec for dy/dt:

2(8)(dx/dt) = 4(5)(units/sec)

Solving for dx/dt, dx/dt = [20 units/sec]/16, or 5/4 units/sec, or 1.25 units/sec.