Question

(a) If A is the area of a circle with radius r and the circle expands as time passes, find dA/dt in terms of dr/dt.
(b) 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 23 m?

Answer 1

Answer:

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

b) area of the spill increasing when the radius is 23 m

$\frac{dA}{dt} = 46\pi m/s$

Step-by-step explanation:

Explanation:-

a)

Given  'A' is the area of a circle with radius 'r'

The area of the circle  $A = \pi r^{2}$   ..(I)

Differentiating equation (I) with respective to 't'

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

b)

If the radius of the oil spill increases at a constant rate of 1 m/s

Given the radius r= 23m

Area of the spill increasing when the radius is 23 m

$\frac{dA}{dt} = \pi (2)(23) (1) =46\pi m/s$