A particle moving along a hyperbola xy =8. as it reaches the point (4,2), the y-coordinate is decreasing at a rate of 3cm/s. how fast is the x-coordinate of the point changing at that instant.

Question

A particle moving along a hyperbola xy =8. as it reaches the point (4,2), the y-coordinate is decreasing at a rate of 3cm/s. how fast is the x-coordinate of the point changing at that instant.

Accepted Answer

increasing by 6 cm/s.
Since you're looking for rate of change per instant, you need to get the first derivative of the function the point is moving along.
d/dx [xy] = 8
d/dx [y]*x + y*d/dx[x] = 0
y'x + 1y = 0
y'x + y = 0
y'x = -y
y' = -y/x
So the slope of the function at (4,2) is
y' = -2/4
y' = -1/2
The rate of X changing will be this equation that then is solved for X. So
-1/2 = -3/X
-X/2 = -3
X = 6
So at the moment the particle reaches (4,2) the value of the x-coordinate is increasing at a rate of 6 cm/s

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A particle moving along a hyperbola xy =8. as it reaches the point (4,2), the y-coordinate is decreasing at a rate of 3cm/s. how fast is the x-coordinate of the point changing at that instant.

A particle moving along a hyperbola xy =8. as it reaches the point (4,2), the y-coordinate is decreasing at a rate of 3cm/s. how fast is the x-coordinate of the point changing at that instant.