Using implicit differentiation, it is found that:
a) The area is expanding at a rate of 314 m²/min.
b) The area is expanding at a rate of 100.5 m²/min.
The area of a circle of radius r is given by:
\(A = \pi r^2\)
It's implicit derivative in function of time is:
\(\frac{dA}{dt} = 2\pi r \frac{dr}{dt}\)
The radius of a circular oil slick expands at a rate of 2 m/min means that \(\frac{dr]{dt} = 2\), thus:
\(\frac{dA}{dt} = 4\pi r\)
Item a:
Radius of 25 m, thus \(r = 25\).
The rate that the area is expanding is:
\(\frac{dA}{dt} = 4\pi (25) = 314\)
The area is expanding at a rate of 314 m²/min.
Item b:
Radius starts at 0, expands at a rate of 2 m/min, thus, after 4 minutes, \(r = 8\).
The rate that the area is expanding is:
\(\frac{dA}{dt} = 4\pi (8) = 100.5\)
The area is expanding at a rate of 100.5 m²/min.
A similar problem is given at https://brainly.com/question/24861732
