Question

The Area of a circle increases at a rate of 1 cm^2/s. How fast is the radius changing when the circumference is 2 cm?

Answer 1

The area of a circle:
 A = πr²
dA/dr = 2πr

We are given dA/dt, using the chain rule
dr/dt = dA/dt x dr/dA

dr/dt = 1 x 1/2πr

When circumference = 2 cm:
2 = 2πr
r = 1/π; putting this value of r

dr/dt = 1/2π(1/π)
dr/dt = 1/2 cm/s

The radius is increasing at a rate of 1/2 cm/s.

Answer 2

¹/₂ cm/s

Further explanation

Given:

The area of a circle increases at a rate of 1 cm²/s.

Question:

How fast is the radius changing when the circumference is 2 cm?

The Process:

Step-1: differentiate the area equation of a circle

The area of the circle is $\boxed{ \ A = \pi r^2 \ }$.

Let us differentiate this function with respect to r.

$\boxed{ \ \frac{dA}{dr} = 2 \pi r \ }$

Step-2: using the chain rule

Let us use the chain rule for composite functions.

$\boxed{ \ \frac{dA}{dr} = 2 \pi r \ }$

Hence $\boxed{ \ \frac{dA}{dt} \cdot \frac{dt}{dr} = 2 \pi r \ }$

Step-3:  find out how fast is the radius changing when the circumference is 2 cm

We knew that,

• $\boxed{ \ \frac{dA}{dt} = 1 \ cm^2/s \ }$
• the circumference is 2πr = 2 cm

Let us substitute in the equation from the previous chain rule.

$\boxed{ \ 1 \cdot \frac{dt}{dr} = 2 \ }$

Multiply both sides by dr/dt and ¹/₂.

$\boxed{ \ 1 = 2 \cdot \frac{dr}{dt} \ }$

Multiply both sides by ¹/₂.

$\boxed{ \ \frac{dr}{dt} = \frac{1}{2} \ }$

Thus, we get how fast the radius changing when the circumference is 2 cm, which is ¹/₂ cm/s.

Learn more

1. Using the product rule  brainly.com/question/1578252  
2. The derivatives of the composite function  brainly.com/question/6013189  
3. What is the general form of the equation of the given circle with center A(-3,12) and the radius is 5? brainly.com/question/1506955