Question

related rates: problem 3 (1 point) 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 cm³, 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? cm³/min preview my answers submit answers you have attempted this problem 0 times. you have unlimited attempts remaining.

Explanation:

Step1: Find the value of constant C

Given $P = 150$ kPa and $V=600$ $cm^{3}$, from $PV = C$, we have $C=150\times600 = 90000$. So the equation is $PV = 90000$.

Step2: Differentiate the equation with respect to time t

Using the product - rule $(uv)^\prime=u^\prime v + uv^\prime$, where $u = P$ and $v = V$. Differentiating $PV = 90000$ with respect to $t$ gives $\frac{dP}{dt}V+P\frac{dV}{dt}=0$.

Step3: Substitute the known values

We know that $\frac{dP}{dt}=20$ kPa/min, $P = 150$ kPa and $V = 600$ $cm^{3}$. Substituting these values into $\frac{dP}{dt}V+P\frac{dV}{dt}=0$:
$20\times600+150\frac{dV}{dt}=0$.

Step4: Solve for $\frac{dV}{dt}$

First, simplify the left - hand side of the equation: $12000 + 150\frac{dV}{dt}=0$. Then, isolate $\frac{dV}{dt}$:
$150\frac{dV}{dt}=- 12000$.
$\frac{dV}{dt}=\frac{-12000}{150}=-80$ $cm^{3}/min$.

Answer:

$80$