oil spilled from a ruptured tanker

Question

Oil spilled from a ruptured tanker spreads in a circle whose area increases at a constant rate of 9 mi^2/hr. How rapidly is radius of the spill increasing when the area is 9 mi^2?

Answer 1

 

  Area = pi * radius^2 where radius and Area are both functions of time. I use 22/7 for pi.

 

 Then A = 22/7* R^2

 

When the area is 9 ,  the radius is  9 = 22/7*R^2

 

                                                    63/22 = R^2

 

                                                   (63 * 22)/(22 * 22) = R^2  <--- rationalizing the denominator before square root

 

                                                    3*square-root(154)/22 = R                                              

 

                                         I will abbreviate square-root(154) as rad154 henceforth

 

Now the derivative:

 

dA/dt = 44/7*R * dR/dt

 

 

Plugging in the given or calculated values:

 9  = 44/7 * 3*rad154/22 * dR/dt

 

9 * 22 * 7

--------------------   =  dR/dt

 44 * 3 * rad154

 

 

3 * 7

-------------  =   dR/dt

2 * rad154

 

 

21 * rad154

---------------   <--- rationalizing again

  308

 

3 * rad154

-------------  = .84611 miles per hour

      44