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Question

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


Solution

we need to find coordinates of point where $$\dfrac{dy}{dt}=2\dfrac{dx}{dt}$$
$$6y=x^3+2$$
Differentiating on both sides
$$6\dfrac{dy}{dt}=3x^2\dfrac{dx}{dt}$$
$$6\times 2\dfrac{dx}{dt}=3x^2\dfrac{dx}{dt}$$
$$x^2=4$$
$$x=\pm 2$$
If $$x=2\Rightarrow y=\dfrac{5}{3}$$
if $$x=-2\Rightarrow y=-1$$

Mathematics

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