Question

The radius of a circle increases at the rate of 2 cm per second. Find the rate at which its area and circumference increase at the instant when the radius is 7cm.

Solve using Derivative function.​

Answer 1

Given :

The radius of a circle increases at the rate of 2 cm per second.

$\sf\implies\dfrac{dr}{dt}=2cms^{-1}$

To Find :

The rate at which its area and circumference increase at the instant when the radius is 7cm.

Solution :

We know that

Area of a circle

$\sf\:A=\pi\:r^2$

Now , differentiate it with respect to t

$\sf\implies\dfrac{dA}{dt}=\pi\times2r\times\dfrac{dr}{dt}$

$\sf\implies\dfrac{dA}{dt}=2r\:\pi\times\dfrac{dr}{dt}$

$\sf\implies\dfrac{dA}{dt}|r=7cm|\:=14\pi\times\dfrac{dr}{dt}$

$\sf\implies\dfrac{dA}{dt}|r=7cm|\:=14\pi\times2$

$\sf\implies\dfrac{dA}{dt}|r=7cm|\:=28\pi\:cm^2\:s^{-1}$

Now ,

Circumference of a circle

$\sf\:C=2\pi\:r$

$\sf\implies\dfrac{dC}{dt}=2\pi\times\dfrac{dr}{dt}$

$\sf\implies\dfrac{dC}{dt}|r=7cm|\:=4\pi\:cm\:s^{-1}$

Answer 2

Given thαt,

• The radius of a circle increases at the rate of 2 cm per second.
• $\displaystyle{\sf{\dfrac{dr}{dt}=2cms^{-1}}}$

❍ We need to find the rate at which its area and circumference increase at the instant when the radius is 7cm.

__________

To do so,

• We'll find the area first.

Areα of the circle = $\large{\boxed{\sf{\orange{\pi r^{2}}}}}$

• After that,differentiate it with respect to t.

$\sf\implies\dfrac{dA}{dt}=\pi\times2r\times\dfrac{dr}{dt}$

$\sf\implies\dfrac{dA}{dt}=2r\:\pi\times\dfrac{dr}{dt}$

$\sf\implies\dfrac{dA}{dt}|r=7cm|\:=14\pi\times\dfrac{dr}{dt}$

$\sf\implies\dfrac{dA}{dt}|r=7cm|\:=14\pi\times2$

$\implies{\sf{\dfrac{dA}{dt}|r=7cm|\:}}$

$\large\implies{\boxed{\sf{\pink{28\pi\:cm^2\:s^{-1}}}}}$

Now ,

Circumference of a circle = $\displaystyle{\boxed{\sf{\orange{2 \pi r^{2}}}}}$

$\sf\implies\dfrac{dC}{dt}=2\pi\times\dfrac{dr}{dt}$

$\implies{\sf{\dfrac{dC}{dt}|r=7cm|\:}}$

$\large\implies{\boxed{\sf{\red{4\pi\:cm\:s^{-1}}}}}$

• Therefore, the rate at which its area and circumference increase at the instant when the radius is 7cm is $\displaystyle{\sf{4\pi cm s^{-1}}}$.

And we are done! ✔

_________

More about the topic!

Rate of change of quanties

let y = f(x)

If the change in one quantity y varies with a another quantity x , f'(x) denotes the rate of change of y with respect to x.

__________________________