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Suppose that the oil spill from the damaged hull of a ship forms a circular slick whose thickness is uniformly 1 millimeter and whose radius is increasing at a rate of 20 kilometers per day. At the instant when the radius of the oil slick is 15 kilometers: (a) Determine the rate at which the spill is flowing out of the ship. (b) What effect does a 12 hour time delay in plugging the leak have on the spill? (Note: the volume of a circular disc of radius $r$ and thickness $d$ is $\pi r^{2} d,$ and 1 millimeter $=10^{-6}$ kilometer.)

(a) $3 \pi / 5000$ cubic $\mathrm{km} / \mathrm{day}$(b) It will increase by $10 \mathrm{km}$

Calculus 1 / AB

Chapter 2

An Introduction to Calculus

Section 10

Related Rates

Derivatives

Campbell University

Harvey Mudd College

University of Nottingham

Idaho State University

Lectures

04:40

In mathematics, a derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's velocity. The concept of a derivative developed as a way to measure the steepness of a curve; the concept was ultimately generalized and now "derivative" is often used to refer to the relationship between two variables, independent and dependent, and to various related notions, such as the differential.

30:01

In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (the rate of change of the value of the function). If the derivative of a function at a chosen input value equals a constant value, the function is said to be a constant function. In this case the derivative itself is the constant of the function, and is called the constant of integration.

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for this problem. We're while looking at an oil spill. So this oil spill ISS circular. So we're going to start withdrawing a circle. Pictures are a great way to see what's happening. I'm gonna label the are with a variable and not a number, because it's changing. The more oil that goes into here, the larger the slick is. So let's write down what we know. First, we know that the thickness is one millimeter. So we'll say that the height of this spill is one millimeter. Okay, universal across the board. Next, we know that the radius is increasing. Okay, Well, that's D r d t. It's increasing. So this is a positive not a negative, uh, number here, and it's increasing at a rate of 20 kilometers per day. Now, all of us already, I can see we have an issue here. The radius is being shown in, uh, Kilometers. My height is millimeters. So we're going to rewrite this height, so we have the same units all the way across. It's very important to make sure we've got consistent units were told that a millimeter is one is sorry is 10 to the negative sixth kilometers so I can write that height is 10 to the negative six kilometers instead of one millimeter. Now, the question we're being asked for part a is what is the rate at which the spill is flowing out. So here's my unknown D v D t. How fast is the volume of this spill changing? Okay, at a certain point in time. And that is when the radius is 15 kilometers again, this doesn't go on. Our picture is just a snapshot. That is the point, though, that we're going to evaluate our equation. So let's begin Now that we have our picture, we've written down all the information we know now is the time to write an equation. How do these variables relate to each other? Well, we have a very, very, very thin cylinder, but it is a cylinder is a circle with a height, So the area ours are the volume of a cylinder is pi r squared H Well, in this case, H isn't really a variable. We know it's a uniformed, constant, unchanging number. So what I really have here is 10 to the negative six pie R squared, that is two variables viene are And if you look at our rates, we know one of them and we want the other. So we are ready to solve this. Let's take the derivative on the left. We have DVD T on the right. Bring that to downs was two times 10 to the negative. Six pi times are D R D t. Now let's look at our snapshot moment. D V D t is what we're trying to find, so we should be able to substitute for everything else. Two times 10 to the negative six times pi r At my point in time is 15 d r d t is 20 so we can simplify this a bit. I have 15 times, 20 times, too. That's going to be 600 times 10 to the negative six pie so we can write this in a couple of different ways. I can, um, put those together and say that six times 10 to the negative fourth pie is kind of putting it in scientific notation. Or if we wanted to, I could write this as a fraction so I could take this and say it's 600 pie over 10 to the negative six, which is 123456 Syrahs, Cancel some of these out. And that gives me 6/10. Yep. 123456 six Over 10,000 or three pi over 5000. Sorry, I was off by a zero here looking at it. But yes, it could be so we couldn't do it either of two ways. Some people like to see it like this. That makes a little more sense. Some people like to see it as a fraction. In either case, it's the same number. So that is how the volume is changing at this point in time. Now the other question is, what's the effect of a 12 hour time delay in plugging the leak of the spill? Well, what's happening in 12 hours? Well, in 12 hours off the how is the radius changing? It's changing at 20 kilometers per day, So half a half of a day 12 hours is half of that distance. The radius will increase by 10 kilometers. So the longer we let this go, the more and more oil is going to go out in our spill. So we want to catch this as early as possible toe Let the minimum amount of oil enter the ocean

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