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The radius of a spherical ball is increasing at a rate of $ 2 cm/min. $ At what rate is the surface area of the ball increasing when the radius is $ 8 cm? $
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Calculus 1 / AB
Harvey Mudd College
University of Nottingham
Idaho State University
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.
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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we're told that the radius of hysterical ball is increasing at a rate of two centimeters per minute. So if we call the Radius, are in time T we have that d R D t is equal to positive, too, and were asked at what rate the surface area of the ball was increasing and the radius is eight centimeters. So are the radius is eight to determine the rate at which the surface area goal is increasing. First, it's fine what the surface area is in terms of the radius of the spiritual ball. So we know that the surface area of a bull this is four pi r squared and therefore we have the rate of change of the surface area with respective time. This is the SPT and by the chain rule. This is eight pi r times d r d t. And so, in the context of this problem, when R is equal to eight and be our duty is to this is a pie times eight times two, which is 128th pie. I don't know. It's from three, and this is the rate of change of the surface area surface area is in this case, square centimeters and the time is in minutes. So this is 128 high square, centimeters per minute. I'm so mad.
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