SOLUTION: A point is moving along the circle x2+y2= 25 in the first quadrant in such a way that its x coordinate changes at the rate of 2cm/sec. how fast is its y coordinate changing as the

Question 650829: A point is moving along the circle x2+y2= 25 in the first quadrant in such a way that its x coordinate changes at the rate of 2cm/sec. how fast is its y coordinate changing as the point passes through (3,4)?
Answer by Edwin McCravy(20086) (Show Source): You can put this solution on YOUR website!
A point is moving along the circle x� + y� = 25

>>...x coordinate changes at the rate of 2cm/sec...<<

The vector pointing right is  = 2 cm/sec (right because it's
positive. The vector pointing down is  = ? cm/sec.
So the point is moving clockwise, and since the point is moving downard
toward the x-axis,  vector is pointing downward, so we can
expect its speed to be to be negative.

>>how fast is its y=coordinate changing as the point passes through (3,4)?<<

We want to know:  = ? when x=3 and y=4

CAUTION:
x and y are varying as the point is moving before it reaches (3,4),
so DO NOT substitute for x or y until after we have taken the derivative.
Only then do we freeze the motion at the point (3,4)

                  x� + y� = 25

            2x + 2y = 0

Divide through by 2

              x + y = 0

                     y =  -x

                       =  �

Now finally we can freeze the motion by substituting x=3, y=4,  = 2

                       =  �(2)

                       =  

                       =  -1.5 cm/sec

That means that at the instant when the point passes through (3,4) 
it is falling toward the x-axis at a rate of 1.5 cm/sec.

Edwin

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