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(a) If $ A $ is the area of a circle with radius $ r $ and the circle expands as time passes, find $ dA/dt $ in terms of $ dr/dt. $(b) Suppose oil spills from a ruptured tanker and spreads in a circular pattern. If the radius of the oil spills increases at a constant rate of $ 1 m/s $, how fast is the area of the spill increasing when the radius is $ 30 m? $

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04:17

Alex Lee

Calculus 1 / AB

Chapter 3

Differentiation Rules

Section 9

Related Rates

Derivatives

Differentiation

Missouri State University

Oregon State University

Harvey Mudd College

University of Nottingham

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.

44:57

In mathematics, a differentiation rule is a rule for computing the derivative of a function in one variable. Many differentiation rules can be expressed as a product rule.

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(a) If $ A $ is the area o…

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(a) If A is the area of a …

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If A is the area of a circ…

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(a) If $A$ is the area of …

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The radius of a circular o…

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Yes, in part A, we're told, is the area of the circle of radius R and circle. It stands as time passes and ratifying d a d t. In terms of DRC's well, it is. The area of a circle of radius R A is equal to high on swear and therefore thinking there isn't on both sides with respective t. We have that already today, with respect to t is equal to buy the chain rule. We have two times high art times. Derivatives are with respect to teach and so we have DVT in terms of the Arctic highest, then in part B or call to suppose that oil spills from a ruptured tanker and spreads in a circular Tatham. We're told the radius of the oil spills increases at a constant rate of one meters per second and rest how fast the area of the still is increasing and the radius is 30 m. In other words, this is part A except for yeah, changing the radius drd t. This is one million per second and your radius is 30 m always. And so we have that Yes, the rate at which the area of the spill is changing d a d t. Well, this is two times pi times 30 times one, which is 60 pie and the human. It's are well, the area is in square meters and the time is in second. So square meters per second to work it's

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