Question

An oil tank is leaking into a lake at a rate of 0.1 m3/day. The oil slick forms a semicircular disk whose center is at the leak site, and it has a thickness of 10−6 meters. How rapidly is the slick expanding when 0.5 cubic meters of oil have been leaked?

Answer 1

Answer:

17.9  m/s

Step-by-step explanation:

Volume of the slick =  0.5 x π r² h--------------------------------- (1)

Where r = radius of slick

           h = thickness of slick, 10⁻⁶m

If 0.5m³ of oil leaked, then the radius of the semicircular slick can be calculated from equation (1)

V  =  0.5 x π r² h

0.5=  0.5 x  π x  r² x 10⁻⁶

r²   =  10⁶/ π

r =     10³/√π

dV/dt =   πrh dr/dt  + 0.5π r² dh/dt----------------------------------- (2)

Asumming the film thickness is constant , equation (2) becomes

dV/dt =   πrh dr/dt-------------------------------- (3)

dV/dt = 0.1m³/day

r=  10³/√π

dr/dt= rate of expansion of the slick

Substituting  into (3);

0.1 =  π x 10³/√π x 10⁻⁶ x dr/dt

dr/dt = 0.1  x 10⁶/ ( π x 10³/√π)

       =  17.9479 m/s

       ≅  17.9  m/s