Can someone help me with a related rates problem please?

Question

An oil slick on a lake is surrounded by a floating circular containment boom. As the boom is pulled in, the circular containment area shrinks. If the radius of the area decreases at a constant rate of 5 m/min, at what rate is the containment area shrinking when the containment area has a diameter of 60 m? please give exact answer

Answer 1

An oil slick on a lake is surrounded by a floating circular containment boom. As the boom is pulled in, the circular containment area shrinks. If the radius of the area decreases at a constant rate of 5 m/min, at what rate is the containment area shrinking when the containment area has a diameter of 60 m? please give exact answer

 

The area of a circle is:

 

A = (pi)r²

 

The rate at which the area decreases as the radius decreases is the first derivative of A wrt time (t):

 

A' = dA/dt = (pi)*(dr²/dt)

 

Use the chain rule to evaluate d(r²)/dt = (2r)*(dr/dt) = (2r)(-5) = -5d where d= the diameter of the containment area and dr/dt = -5, the rate that the radius is decreasing

 

A' = (-5)(pi)(d)

 

When the containment area has a diameter of d = 60m

 

A' = (-5)(pi)(60) = -300(pi) m²/min ≅ -942.48 m²/min