Expert: Paul Klarreich - 2/5/2008

Question
I have been studying related rates but this question confuses me as to what formula to use.

A tanker is spilling oil into the water resulting in an oil slick that is close to circular.  At the time that the slick's diameter is growing at the rate of 7 m/min, the diameter of is 150 meters.  At what rate is the area of the oil slick increasing?
The multiple choice answers are:
a. 235.619 m squared/ min
b. 525.000 m squared/ min
c. 1649.335 m squared/ min
d. 3298.672 m squared/ min

Thank you.

Answer
Questioner:   Aly
Category:  Calculus
Private:  No
 
Subject:  Related Rates Question
Question:  I have been studying related rates but this question confuses me as to what formula to use.

A tanker is spilling oil into the water resulting in an oil slick that is close to circular.  At the time that the slick's diameter is growing at the rate of 7 m/min, the diameter of [OF WHAT?] is 150 meters.  At what rate is the area of the oil slick increasing?

The multiple choice answers are:

a. 235.619 m squared/ min
b. 525.000 m squared/ min
c. 1649.335 m squared/ min
d. 3298.672 m squared/ min

Thank you.
..................................
Hi, Aly,

If the problem has to do with the area of a circle, you use the formula for the area of a circle.  Is that all you needed?

But is this your first attempt at R-R problems?  If so, the scheme is something like this:

1. Identify the variables in the problem -- the things that change.  Give them names.

2. Write their rates of change as derivatives WITH RESPECT TO time.  Note which are known and which is to be found.

3. Determine a relationship (yes, it is called 'related rates' for a reason) between the variables.  Use a diagram, use your life experience, use your general knowledge and brilliance, do whatever you have to.  This is the key step.

4. Now differentiate implicitly, then substitute the known quantities and rates, and solve for the unknown rate.
.............................
A tanker is spilling oil into the water resulting in an oil slick that is close to circular.  At the time that the slick's diameter is growing at the rate of 7 m/min, the diameter of [OF WHAT?] is 150 meters.  At what rate is the area of the oil slick increasing?

Organize the work:

Let  D = the diameter, and
    r = the radius, of the slick.
    A = the area of the slick.

dD/dt = the rate of increase of Diameter. Given as 7 m/min
dA/dt = ....................... Area.  TO BE FOUND.
dr/dt = ....................... radius.

and, of course, you have these rules:

r = D/2,
dr/dt = 1/2 dD/dt = 1/2(7) = 7/2

Now the relation:

A = pi r^2

Diff:
dA/dt = 2 pi r dr/dt

Subst:  r = 75 m, dr/dt = 7/2

dA/dt = 2 pi (75)(7/2)
dA/dt = pi (75)(7)
Now you can do the arithmetic and make the choice.