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?

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.