1) Oil is leaking at a rate of R(t) = 2000e-0.2t
, where t is measured in hours, how much oil has spilled in 10 hours?
We want to end up in gallons, so we multiply gallons per hour (which we are given) by some change in time. We want to find the total sum over the ten hours.
*When we are given a rate, such as the one above, we are trying to find a unit related to the rate. I.e. (gallons/hour)x(hour) = gallons.*
A) Density of an oil slick on a body of water is defined by
We have Kg/m2 and we want to get kg, so we have to get rid of the m2
We cut a piece from the oil slick. If we drill down, the density of the oil slick is the same throughout.
Here we have cut a piece out of the oil slick. The ends of the strips (triangles) become smaller. So we take the triangle and move it to one side to form a rectangle. The area of the rectangle is what we need to find. Going back to our equation we can now get rid of m2
With our our information and our missing piece found, we can create an integral to solve for the mass of the oil slick. We take the limit of n as it approaches infinite and it becomes smaller. We can then take the sum, where n equals one. p(a) is our given equation so we multiply that by the rectangle piece we cut out, Am2. Then we get the integral from 0 to 1000 meters. 100/(1+r2) is kg/m2 multiplied by 2pi r dr m2, which will give us the mass of the oil slick 1000 m out. 2pi r is multiplied by dr because dr becomes infinitely smaller.
*check the slide for march 19 to view the solution to the integral*
B)What is the smallest radius that contains 75% of the oil slicks mass?
With our solution from A, all we do is take seventy percent of the value we found. We can then pull out 200pi. From here we just solve for r, by expanding the bracket notation first.
For the last part of class, we talked about how our physics formula can be derived. We also did a quick question before the bell rang. We were given an amount of cars per kilometer. instead of trying to cancel out a unit area we had to cancel a length to get the number of cars.
URGH, i got a cold.... next scribe is Suzanne... pre test tmrw?