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Boyle's law for gasses states that $P V=$ constant, where $P$ is the pressure in pounds per square inch (psi) and $V$ is the volume of the gas. For a particular gas it is known that when the pressure is 10,000 psi, its volume is 4 cubic inches. If the volume is increasing at 8 cubic inches per second, find the rate at which the pressure is changing when the pressure is 20,000 psi.

-80,000 psi/sec

Calculus 1 / AB

Chapter 2

An Introduction to Calculus

Section 10

Related Rates

Derivatives

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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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Boyle's law for enclo…

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As discussed in this secti…

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Boyle's Law Boyle…

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Boyle's Law states th…

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Boyle's Law. The volu…

this problem starts by reminding us what Boyle's law for gas is. Is Boyle's law for gas is, says pressure times. Volume is a constant for a certain gas. So as one goes up, the other one would have to go down preppies. Pressure in pounds per square inch V is the volume, and we're also told we have a particular gas that when the pressure is 10,000 PCs, the volume is four cubic inches. We can use that to find the constant for this particular gas. So the pressure is 10,000 and the volume is four. That gives me a value of 40,000. So for this particular gas, we have the equation. P Times V equals 40,000. That's our equation that we can use. Now. We're going to play around with it a little bit. We're told that the volume is increasing. O R. Volume is changing, so that's a change of volume. With respect to time, the volume is increasing at eight cubic inches per second. It's increasing. That's a positive age cubic inches per second. We want to find the rate at which the pressure is changing, so that's are unknown and we want to know at at the point when the pressure is 20,000. Okay, so that's what we have. We know DVD T. We want DP DT. That's two unknowns. Our equation has to a notes. So we're good. PV equals 40,000. Let's take the derivative. We have a quotient. So that would be the first times the derivative of the second plus the second times, the derivative of the first and the derivative a constant is zero. Okay, let's plug in what we know. Well, I know that the pressure is 20,000 PSE DVD t. We're told it's increasing. That's a positive eight. Now volume we don't know. So I'm going to just leave a little bit of ah, break here for a second. Dp DT is are unknown. So how do we find the how do we find value? Well, fortunately, we have an equation that relates to two. That's the usually are starting point with a related rates problems. How do these rates relate? Well, in this case, for our particular instant, my P, my pressure is 20,000. It always equals 40,000, which means V would have to equal to in this case so at our instant V is too. And now we can solve this. I'm gonna move up here about that. Just a little more room. I have to dp DT. When I prove that over, this could be a negative 160,000 and then I divide by two as a negative. 80,000 p. S. I. And that number makes sense because you can see it's a constant. If one increases, the other has to decrease. In order to keep that balance, make sure that product equals a constant. Since I know my volume is increasing, it makes sense that my pressure would be decreasing.

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