Question

The radius of a circle is increasing at the rate of $0.7cm/s$. What is the rate of increase of its circumference ?

• A. $2\pi cm/s$
• B. $0.7\pi cm/s$
• C. $1.4\pi cm/s$
• D. $Noneofthese$

Answer 1

Answer: (C)

$1.4\pi cm/s$

Let $r$ be the radius of circle and $c$ be the circumference of circle.

Given that,

Radius of a circle is increasing at rate of $0.7cm/sec$

i.e, $\frac{dr}{dt}=0.7cm/sec⟶\left(1\right)$

we need to find rate of change of circumference of circle wrt time.

We need to calculate $\frac{dc}{dt}$

we know that circumference $=2\pi r$

i.e, $c=2\pi r$

Differentiating wrt time

$\Rightarrow \frac{dc}{dt}=\frac{d\left(2\pi r\right)}{dt}$

$\frac{dc}{dt}=2\pi \frac{dr}{dt}$

$\frac{dc}{dt}=2\pi \times 0.7$

$\frac{dc}{dt}=1.4\pi$

since, circumference is in cm $4$ times is in sec

$\Rightarrow \frac{dc}{dt}=1.4\pi \frac{cm}{sec}$

$\frac{dc}{dt}=1.4\pi cm/sec$

Hence, circumference of circle is increasing at rate of $1.4\pi cm/sec$ or $1.4\times 3.14cm/sec=4.396cm/sec.$

Hence, the answer is $1.4\pi cm/sec.$