If radius of a circle is decreasing at the rate of 3cm/s find rate of change of circumference when r=2
C = 2πr
dC/dt = 2π dr/dt
Now just plug and chug. You will note that dC/dt does not depend on r.
OOBleck is correct, but just to be clear (and this answer is about a year late).
So plugging and chugging... 2 Pi times -3 cm/sec (negative since it's decreasing) Gives -6 Pi ... this is the "rate of change" of C. it is -6 Pi at r=2, and -6 Pi at any other r. so, it is independent of r. However, that does not tell us what the circumference actually is at any r. But without the derivative, we know C = 2 Pi r. at r=2, C = 4 Pi cm.
To find the rate of change of the circumference when the radius is 2 cm, we will differentiate the formula for the circumference of a circle with respect to time.
The formula for the circumference of a circle is C = 2πr, where C is the circumference and r is the radius.
Differentiating both sides of the equation with respect to time (t), we get:
dC/dt = 2π(dr/dt)
Since we are given that the radius is decreasing at a rate of 3 cm/s, we can substitute dr/dt with -3 (negative sign indicates a decrease) and r with 2:
dC/dt = 2π(-3)
= -6π
Therefore, the rate of change of the circumference when the radius is 2 cm is approximately -6π cm/s.
To find the rate of change of the circumference when the radius is 2 cm, we need to use the formula for the circumference of a circle, which is given by C = 2πr, where C is the circumference and r is the radius.
We are given that the rate of change of the radius is 3 cm/s, so we need to find the rate of change of the circumference with respect to time. This can be done by taking the derivative of the circumference formula with respect to time.
Taking the derivative of the circumference formula C = 2πr, we get dC/dt = 2π(dr/dt).
Since we know that dr/dt = -3 cm/s (negative because the radius is decreasing), we can substitute this value into the derivative expression to find the rate of change of the circumference.
dC/dt = 2π(-3) = -6π cm/s.
Therefore, when the radius is 2 cm and it is decreasing at a rate of 3 cm/s, the rate of change of the circumference is -6π cm/s.