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 34 m?

Answer 1

The solution would be like this for this specific problem:

 

Given:

 

Oil spill radius constant rate increase = 1 m/s

r = 34m

 

Let the area of the spill be A and then let its radius be r. 
Then A = π r². 

 

Differentiating with respect to t: 
dA / dt = 2π r dr / dt. 

 

Substituting r = 34 and dr / dt = 1: 

 

dA/dt = 2 π * r * dr / dt

 

dA/dt = 2π * 34 * 1 = 68π = 214 m²/s, to 3 significant figures. 

 

So, given that the radius is 34m, then the area of the spill is increasing at 214 m²/s.