Question

Suppose oil spills from a ruptured tanker and spreads in a circular pattern. If the radius of the oil spill increases at a constant rate of 1 m/s, how fast is the area of the spill increasing when the radius is 30 m?

Answer 1

Answer:

$\frac{dA}{dt} = 188.5 m^2/s$

Explanation:

As we know that area of the circle at any instant of time is given as

$A = \pi r^2$

now in order to find the rate of change in area we will have

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

here we know that

rate of change of radius is given as

$\frac{dr}{dt}= 1 m/s$

radius of the circle is given as

$r = 30 m$

now we have

$\frac{dA}{dt} = 2\pi (30)(1)$

$\frac{dA}{dt} = 60\pi$

$\frac{dA}{dt} = 188.5 m^2/s$