Question

The area of a circular oil slick on the surface of the sea is increasing at a rate of [150m^2/s] how fast is the radius changing when:

a) the radius is 25 m

b) the area is [1000 m^2]

Answer 1

Answer with Step-by-step explanation:

We are given that

`(dA)/(dt)=150m^2/s`

a.Radius =25 m

We have to find rate of change of radius.

We know that

Area of circle,
`A=\pi r^2`

Differentiate w.r.t time

`(dA)/(dt)=2\pi r(dr)/(dt)`

Substitute the values then we get

`150=2\pi (25)(dr)/(dt)`

`(dr)/(dt)=(150)/(50\pi)=(3)/(\pi) m/s`

`(dr)/(dt)=(3)/(\pi) m/s`

b.A=
`1000m^2`

Substitute the values then we get

`1000=\pi r^2`

`\pi=3.14`

`r^2=(1000)/(3.14)`

`r=\sqrt{(1000)/(3.14)}=17.8m`

`(dA)/(dr)=2\pi r(dr)/(dt)`

Substitute the values then we get

`150=2\pi(17.8)(dr)/(dt)`

`(dr)/(dt)=(150)/(2\pi(17.8))=(150)/(35.6\pi)`m/s

`(dr)/(dt)=(75)/(17.8\pi)`m/s