A particle is moving along the curve y = x √ x, x ≥ 0. At time t its coordinates are (x(t), y(t)). Find the po

A particle is moving along the curve y = x
√
x, x ≥ 0. At
time t its coordinates are (x(t), y(t)). Find the points on the
curve, if any, at which both coordinates are changing at the
same rate. please explain and the steps

Let's begin by rewriting

y = x √ x ; x ≥ 0

as

y = x^(3/2) ; x ≥ 0

Using the change rule

dy/dx = dy/dt{dt/dx}

For both of the coordinates to be changing at the same rate:

dy/dt = dx/dt

or

dy/dt(dt/dx) = 1

Therefore;

dy/dx = 1

d[x^(3/2)]/dx = (3/2)x^(1/2)

(3/2)x^(1/2) = 1

x^(1/2) = 2/3

x = 4/9

y = x^(3/2)

y = (4/9)^(3/2) = (2/3)^3 = 8/27

The point is (4/9, 8/27)

You are welcome.

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