A storm at sea has damaged an oil-rig. Oil spills from the rupture at the constant rate of 47 ft^3/min, forming a slick that is roughly circular in shape and 5 inches thick.

Question

(a) When the radius is , the radius of the slick is increasing by _____ feet per minute, rounded to the nearest hundredth. (Hint: Make sure the units match for all quantities before plugging in numbers).

(b) If the radius of the slick is increasing at a rate of 0.5 ft/min when the flow stops, which is assumed to be instant rather than gradual, the total volume of oil that spilled onto the sea is _____ cubic feet, rounded to the nearest hundredth.

Answer 1

(a) dV/dt = 47 ft³/min (Looking at the units also gives us a hint of what quantity could it be)

h= 5/12 ft and r = 23 ft

As, V =pi*r*r*h

Differentiating the above equation of V wrt t

dV/dt = pi*2*r*(dr/dt)*h

Plugging in the values

47 = pi*2*23*(dr/dt)*5/12

dr/dt = 0.78 ft/min

(b) dr/dt = 0.5 ft/min

dV/dt = 47 ft³/min

As, V =pi*r*r*h

Differentiating the above equation of V wrt t to solve for r this time

dV/dt = pi*2*r*(dr/dt)*h

Plugging in the values

47 = pi*2*r*(0.5)*5/12

r=35.91 ft

Plugging the values of r and h in the formula for V

V = pi*(35.91)²(5/12)

V=1687.99 ft³/min