A tanker that ran aground is leaking oil that forms a circular slick about 0.2 foot thick. To estimate the rate dV/dt (in cubic feet per minute) at which oil is leaking from the tanker, it was found that the radius of the slick was increasing at 0.31 foot per minute (dR/dtequals0.31) when the radius R was 400 feet. Find dV/dt, using pi almost equals 3.14 .

The value of dV/dt using pi almost equal to 3.14 gives; dV/dt = 155.82 ft³/min

dV/dt = 155.82 ft³/min

We are given;

Radius; R = 400 ft

Height; h = 0.2 ft

Rate of Increase of radius; dr/dt = 0.31 ft/min

Formula for Volume of a cylinder is;

V = πr²h

Differentiation of both sides of V and r with respect to t gives;

dV/dt = 2πrh(dr/dt)

Plugging in the relevant values gives;

dV/dt = 2π × 400 × 0.2 × 0.31

dV/dt = 155.82 ft³/min

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Step-by-step explanation:

Since it is circular slick, it's volume can be modeled as,

Where R is the radius in feet and h is the thickness of slick.

taking derivative of the above equation with respect to time yields,

Where the rate of change of radius of the slick (dR/dt) is given,

Therefore, the rate of change of volume is 155.82 cubic feet per minute.