Question

a circular oil slick is expanding at a rate of 2m^2/h. Find the rate at which it's diameter is expanding when it's radius is 1.5m

Answer 1

Answer:  $\frac{4\pi}{3} \text{ meter per hour}$

Step-by-step explanation:

The circular oil slick is expanding at a rate of  $2 m^2/h$

Let A be the area of the circular oil slick,

So, the changes in  A with respect to time (t),

$\frac{dA}{dt} = 2$

$\frac{d(\pi r^2)}{dt} = 2$

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

$\frac{dr}{dt} = \frac{1}{\pi r}$  

Also, the change in diameter with respect to time(t),

$\frac{d}{dt} (2 r) = 2 \frac{dr}{dt}$

$\frac{d}{dt} (2 r) = 2 \times \frac{1}{\pi r}$

$\frac{d}{dt} (2 r) = \frac{2}{\pi r}$

For r = 1.5 m,

$\frac{d}{dt} ( 2 r)]_{r=1.5} = \frac{2}{\pi \times 1.5}=\frac{20}{\pi \times 15}=\frac{4\pi }{3}\text{ meter per hour}$