Expert: Paul Klarreich - 2/4/2007

Question
So here's the problem, I have been given this problem, and have no idea how to apply this formula.  i've looked up this  type of problem, and have yet to find any remotely similar type problem where this formula is applied.....

(below is the problem)

Suppose there is an oil spill caused by a sinking tanker in the Pacific Ocean.  The oil surfaces continuously, and spreads in a more-or-less circular pattern.  If the spread of the oil spill is modeled by a circle, we can find the circumference of the circle to determine how far away we will have to put the booms to contain the oil, depending on how long it takes us to get the booms to the spill.  Let us suppose that the radius of the circle is changing by one inch per hour.   

Since the oil is surfacing continually, the function that will give a relationship between the time and the radius of the oil spill is given by the expression:

     f(t) = ert  (that should say e to the rt power, not e*r*t)

where:   f(t) is the length of the radius at time t;  t is measured in hours with t = 0 corresponding to the time the spill began;  r is the rate of growth of the radius and is equal to 1” per hour.

Assume that we find out about the spill immediately after it occurs.

a)  If it takes us 10 hours to get the manpower, ships and materials to the spill, how large an area will we have to contain?  

b)  Equally important, how much “boom” will we need?  

Answer
Questioner:   Ryan
Category:  Calculus
Private:  no
 
Subject:  Radius question
Question:  So here's the problem, I have been given this problem, and have no idea how to apply this formula.  i've looked up this  type of problem, and have yet to find any remotely similar type problem where this formula is applied.....

>> OK, OK, what's the problem?

(below is the problem)

Suppose there is an oil spill caused by a sinking tanker in the Pacific Ocean.  
The oil surfaces continuously, and spreads in a more-or-less circular pattern.  

>> I think we'll assume it IS circular.

If the spread of the oil spill is modeled by a circle, we can find the circumference of the circle to determine how far away we will have to put the booms to contain the oil, depending on how long it takes us to get the booms to

the spill.  Let us suppose that the radius of the circle is changing by one inch per hour.   

Since the oil is surfacing continually, the function that will give a relationship between the time and the radius of the oil spill is given by the expression:

f(t) = e^vt

where:   f(t) is the length of the radius at time t;  t is measured in hours with t = 0 corresponding to the time the spill began;  v is the rate of growth of the radius, r, and is equal to 1” per hour.

Assume that we find out about the spill immediately after it occurs.

>> You mean, when  t = 0?

a)  If it takes us 10 hours to get the manpower, ships and materials to the spill, how large an area will we have to contain?  

b)  Equally important, how much “boom” will we need?  
................................................
Hi, Ryan,

I don't think your conditions are consistent.

I changed a bit of your notation:

1. I set  'r' as the length of the radius, and
2.  v = dr/dt, the rate of increase.

First you say that  r = e^vt, but that  v = 1"/hour.  Is that 1"/hour a constant, or is it the value at  t = 10?  

There is a difference, of course, because if  dr/dt = 1, a constant, then  r = t + C, and we can assume that the C = 0, since at t = 0 the spill has just occurred.

Now I think your problem really should look like this:

Oil spills out of the (sunken?) tanker at a constant rate (which should be given) and spreads in a circular pattern.  If after 10 hours of spillage, the radius of the circle is increasing at the rate of  1 mile/hour, then how big a slick will there be after 10 hours?   [I changed that to 1 mile/hour, because if it's just 1"/hour, why worry?]

If this makes sense, let me know and we can continue.

As to the stuff about booms, I have no idea.