The area of a circle is increasing at the rate of 3cm/sec find the rate change of the circumference when the radius is 2cm
A = πr^2
dA/dt = 2πr dr/dt
3 = 2π*2 dr/dt
dr/dt = 3/(4π)
C = 2πr
dC/dt = 2π dr/dt = 2π * 3/(4π) = 3/2 cm/s
or,
C = 2πr = 2π√(A/π) = 2√(πA)
so, when r=2, A = 4π
dC/dt = (2√π)/(2√A) dA/dt = (2√π)/(2√(4π)) * 3 = 3/2
well duh. This is so much easier. since C = 2πr,
dA/dt = 2πr dr/dt = r * 2π dr/dt = r dC/dt
To find the rate of change of the circumference, we need to use the relationship between the area and circumference of a circle.
The formula for the area of a circle is: A = πr^2, where A is the area and r is the radius.
The rate of change of the area is given as 3 cm/sec, so dA/dt = 3 cm/sec.
We are given the radius r = 2 cm. Now, let's find the rate of change of the circumference.
The formula for the circumference of a circle is: C = 2πr, where C is the circumference and r is the radius.
First, differentiate the equation for the area with respect to time (t):
dA/dt = d(πr^2)/dt
Using the chain rule, we get:
dA/dt = 2πr(dr/dt)
Since dA/dt = 3 cm/sec and r = 2 cm, we can solve for dr/dt:
3 cm/sec = 2π(2)(dr/dt)
dr/dt = 3 cm/sec / (2π(2))
dr/dt = 3 /(4π) cm/sec
So, the rate of change of the radius dr/dt is 3 /(4π) cm/sec.
Now, to find the rate of change of the circumference dC/dt, differentiate the equation for the circumference with respect to time (t):
dC/dt = d(2πr)/dt
Using the chain rule, we get:
dC/dt = 2π(dr/dt)
Since we know that dr/dt = 3 /(4π) cm/sec, we can substitute this into the equation:
dC/dt = 2π(3 /(4π))
Simplifying, we find:
dC/dt = 3/2 cm/sec
Therefore, when the radius is 2 cm, the rate of change of the circumference is 3/2 cm/sec.
To find the rate change of the circumference when the radius is 2 cm, we can use the formula for the area of a circle, A = πr^2, and the formula for the circumference of a circle, C = 2πr.
Given that the area of the circle is increasing at a rate of 3 cm/sec, we can differentiate the area formula with respect to time (t) to find the rate of change of the area:
dA/dt = 2πr(dr/dt)
Here, dA/dt represents the rate of change of the area, and dr/dt represents the rate of change of the radius.
Since we are given the rate of change of the area (3 cm/sec) and the radius (2 cm), we can substitute these values into the equation:
3 cm/sec = 2π(2 cm)(dr/dt)
Now, we can solve for dr/dt:
dr/dt = 3 cm/sec / (2π(2 cm))
= 3 cm/sec / (4π cm)
= 3 / (4π) cm/sec
Therefore, the rate change of the circumference when the radius is 2 cm is 3 / (4π) cm/sec.