Related Rates and an Oil Slick

In summary, a ruptured oil tanker causes a circular oil slick on the surface of the ocean with a radius of 150 meters. The slick is expanding at a rate of 0.1 meter/minute and has a thickness of 0.02 meters. The volume of oil spilled remains fixed. In part A, it is asked how fast the area of the slick is expanding. For part B, it is asked how fast the thickness of the slick is decreasing. While part B is easy to solve using a cylinder, there may be confusion in part A regarding whether the area refers to just the surface area of the top of the slick or the entire cylinder.
  • #1
Jacobpm64
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A ruptured oil tanker causes a circular oil slick on the surface of the ocean. When its radius is 150 meters, the radius of the slick is expanding by 0.1 meter/minute and its thickness is 0.02 meter. At that moment:

(a) How fast is the area of the slick expanding?
(b) The circular slick has the same thickness everywhere, and the volume of oil spilled remains fixed. How fast is the thickness of the slick decreasing?

Part B is easy, I just used a cylinder and figured it out. I'm sure I got it correct.

Part A is where I'm confused. When they say area, I wonder do they mean just the surface area of the top of the slick (a circle), or the surface area of the entire cylinder?

What do you people think?
 
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  • #2
I think they mean just the surface area of the top of the slick.
 

1. What is a related rate in the context of an oil slick?

A related rate in the context of an oil slick refers to the rate at which the size or shape of the oil slick changes over time. This can be affected by various factors such as wind and water currents.

2. How does the rate of change of an oil slick affect the environment?

The rate of change of an oil slick can have severe impacts on the environment. As the oil slick spreads and expands, it can contaminate water sources, harm marine animals and plants, and damage ecosystems.

3. What factors influence the rate of change of an oil slick?

The rate of change of an oil slick can be influenced by various factors such as wind speed and direction, water currents, temperature, and the type and amount of oil spilled. These factors can affect the movement and behavior of the oil slick.

4. How can related rates be used to predict the behavior of an oil slick?

Related rates can be used to mathematically model and predict the behavior of an oil slick. By analyzing the rates of change of different parameters, scientists can make predictions about the movement, size, and shape of the oil slick over time.

5. What are some techniques for mitigating the effects of an oil slick?

There are various techniques for mitigating the effects of an oil slick, including using physical barriers to contain the oil, using dispersants to break up the oil, and using skimmers to remove the oil from the water. Other techniques include bioremediation, which uses microorganisms to break down the oil, and controlled burning, which burns off the oil slick. However, these techniques may have their own environmental consequences and should be carefully evaluated before use.

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