Question

boyles law states that when a sample of gas is compressed at a constant temperature, the pressure $p$ and the volume $v$ satisfy the equation $pv = c$, where $c$ is a constant. suppose that at a certain instant the volume is $600\\ cm^3$, the pressure is $150\\ kpa$, and the pressure is increasing at a rate of $20\\ kpa/min$. at what rate is the volume decreasing at this instant? $cm^3/min$

Explanation:

Step1: Differentiate PV = C with respect to time t

Using the product - rule (uv)'=u'v + uv', where u = P and v = V. We get $P\frac{dV}{dt}+V\frac{dP}{dt}=0$.

Step2: Solve for $\frac{dV}{dt}$

$\frac{dV}{dt}=-\frac{V}{P}\cdot\frac{dP}{dt}$

Step3: Substitute the given values

Given that $V = 600\ cm^{3}$, $P = 150\ kPa$, and $\frac{dP}{dt}=20\ kPa/min$. Then $\frac{dV}{dt}=-\frac{600}{150}\times20$.

Step4: Calculate the result

$\frac{dV}{dt}=- 80\ cm^{3}/min$.

Answer:

The volume is decreasing at a rate of $80\ cm^{3}/min$.