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When air expands adiabatically (without gaining or losing heat), its pressure $ P $ and volume $ V $ are related by the equation $ PV^{1.4} = C, $ where $ C $ is a constant. Suppose that at a certain instant the volume is $ 400 cm^3 $ and the pressure is $ 80 kPa $ and is decreasing at a rate of $ 10 kPa/min. $ At what rate is the volume increasing at this instant?
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02:40
Wen Zheng
Calculus 1 / AB
Chapter 3
Differentiation Rules
Section 9
Related Rates
Derivatives
Differentiation
Harvey Mudd College
Idaho State University
Boston College
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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When air expands adiabatic…
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Pressure In the adiabatic …
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The formula for the adiaba…
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In an adiabatic process (…
Okay, we know pressure and temperature fall in this equation is PV to the 1.4 is equivalent to see. Therefore, we know that we have to determine Devi over DT, right? We're turning the rate in this equation there for recon, right? Devi over. DT is t d p over DT, which we know is negative 10 times V, which is 400 over p which is 80 times 1.4, which I wrote appears the exponents, which is equivalent to 35.7 and the unit is centimeters Q put minute. So we know the volume has to be increasing at this rate.
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