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

17.9 m/s

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