Question

boyles law states that when a sample of gas is compressed at a constant temperature, the pressure p and volume v satisfy the equation pv = c, where c is a constant. suppose that at a certain instant the volume is 700 cm³, the pressure is 140 kpa, and the pressure is increasing at a rate of 10 kpa/min. at what rate (in cm³/min) is the volume decreasing at this instant?

1. -/1 points

brain weight b as a function of body weight w in fish has been modeled by the power function b = 0.007w²/³, where b and w are measured in grams. a model for body weight as a function of body length l (measured in centimeters) is w = 0.12l².⁵³. if, over 10 million years, the average length of a certain species of fish evolved from 15 cm to 24 cm at a constant rate, how fast (in g/yr) was this species brain growing when its average length was 19 cm? (round your answer to four significant figures.)

Explanation:

Step1: Find the value of the constant $C$

Given $P = 140$ kPa and $V=700$ cm³, from $PV = C$, we have $C=140\times700 = 98000$. So the equation is $PV = 98000$.

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 $PV = 98000$ with respect to $t$ gives $P\frac{dV}{dt}+V\frac{dP}{dt}=0$.

Step3: Substitute the known values

We know that $P = 140$ kPa, $\frac{dP}{dt}=10$ kPa/min, and $V = 700$ cm³. Substitute these values into $P\frac{dV}{dt}+V\frac{dP}{dt}=0$:
$140\frac{dV}{dt}+700\times10 = 0$.

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

First, simplify the equation:
$140\frac{dV}{dt}=-7000$.
Then $\frac{dV}{dt}=\frac{-7000}{140}=- 50$ cm³/min.

Answer:

$50$