What Is The Area Covered By The Oil Film In Square Meters In Square Miles

The area of a circular oil spill is increasing at a rate of 21 m/hr^2. At what rate is the radius changing when the radius is 9 meters? And how?

The area of a circular oil spill is increasing at a rate of 21 m[math]^2[/math]/hr. At what rate is the radius changing when the radius is 9 meters? And how?(Note that I’ve corrected an apparent typo in the question; it should be square meters per hour, not meters per squared hour.)We start with the equation for the area of a circle: [math]A=\pi r^2[/math]. Both the area and the radius are changing as time passes — that is, both A and r are functions of t. So, we differentiate both sides of the equation with respect to t:[math]\begin{align*}\frac{d}{dt}\left[A\right] &=\frac{d}{dt}\left[\pi r^2\right] \\\implies \frac{dA}{dt} &=2\pi r\cdot \frac{dr}{dt}&\text{(use the chain rule on the right side)}\end{align*}[/math]We were told that “the area … is increasing at a rate of 21 m[math]^2[/math]/hr”; that is the value of dA/dt. Therefore, when the radius is 9 meters, we have [math]21\frac{\rm m^2}{\rm hr} = 2\pi(9\text{ m})\frac{dr}{dt}[/math]. Solving leads to [math]\frac{dr}{dt}=\frac{7}{6\pi}\,\frac{\rm m}{\rm hr}[/math].Why/how: Part of the beauty (and usefulness) of differentiation is that it is an operation which one can do to both sides of an equation, just like the usual things we do when using algebra to solve an equation (e.g., “subtract 5 from both sides” or “square both sides” or “multiply both sides by [math]x^2+1[/math]").If you start with a valid equation relating two or more variables (like [math]A=\pi r^2[/math]), and differentiate both sides (with respect to t, A, r, or whatever variable you choose), you obtain another valid equation relating the derivatives of those variables.This is the same principle that is used for implicit differentiation.

What do you think of the oil leak in the Gulf of Mexico spilling 300 to 700 barrels of oil per day since 2004?

There’s no way in hell that number is correct. No way in hell that the total volume spilled rivals Macondo, as claimed.Oil spill management training is a requirement of working offshore. I underwent spill training at least annually for approximately 10 years. Spill volumes can be estimated based on the visual appearance of the slick or sheen. More detail is here: http://www.environmentalunit.com...A spill of 300 to 700 bbl/day would necessarily result in a heavy, thick layer of live oil on the water, more like we saw in the BP spill:By contrast, here’s the photo from the WaPo article, which clearly shows a silvery to rainbow sheen, thicknesses measured in micrometers. To quote from the article: “…all that is left of the doomed Taylor platform are rainbow-colored oil slicks that are often visible for miles.”Picture caption: “An aerial image of an oil slick in the Gulf of Mexico, taken on April 28, 2018. (Oscar Garcia-Pineda)”Rainbow sheens are not consistent with spill rates in the hundreds of barrels per day. The moral of the story is that a little bit of oil will cover a deceptively large area on the water.Let’s make up some numbers for illustration: A rainbow sheen is 6 miles long by 300 yards wide on average covers roughly 1 sq mile, or 2.6 sq km. Rainbow sheens have 0.3 to 5.0 cubic meters per sq km. The estimated volume of my hypothetical spill is therefore 0.78 to 13 cubic meters. At 6.3 barrels per cu m, that’s a total of 5 to 80 barrels of oil.Note that the Coast Guard’s estimates of spill volumes range from 0 to 55 barrels per day.Oscar Garcia-Pineda is the source of the 300 to 700 barrel per day estimate. His photo does not document the thick slick that would justify that estimate. Did he mean gallons per day?The WaPo is full of other misstatements and errors. I plan to blog about it. Stay tuned.