Question Type 14: Calculating with proportionality of two variables Exercises

If $y$ is directly proportional to $x^4$ with $k=5$, find the positive value of $x$ when $y=80$.

The pressure $P$ of an ideal gas varies inversely with its volume $V$. If $P=200\,\text{kPa}$ when $V=0.5\,\text{L}$, determine $V$ when $P=80\,\text{kPa}$.

The intensity $I$ of light varies inversely with the square of the distance $d$ from the source. If $I=100$ lux at $d=2\,\text{m}$, find $d$ when $I=25$ lux.

If $y$ varies inversely as the square root of $x$ and $y=6$ when $x=4$, find $y$ when $x=16$.

A population $P$ of bacteria is directly proportional to the square of time $t$. If $P=200$ at $t=3\,\text{h}$, find $P$ at $t=5\,\text{h}$.

Given that $y$ is inversely proportional to $x^2$ and $y = 20$ when $x = 2$, determine the constant of proportionality $k$.

When $y$ is inversely proportional to the cube of $x$, by what factor does $y$ change if $x$ increases from 2 to 4?

According to Newton's law of gravitation, the force $F$ between two masses varies inversely with the square of the distance $r$. If $F=100\,\text{N}$ at $r=2\,\text{m}$, calculate $F$ when $r=5\,\text{m}$.

A quantity $y$ is inversely proportional to $x^p$. When $x=2$, $y=18$ and when $x=3$, $y=8$. Determine the exponent $p$.

If $y$ is inversely proportional to $x$ with constant of proportionality 12, calculate $y$ when $x = 3$.

The variable $y$ is directly proportional to $x^3$ with proportionality constant $2$. Calculate the value of $y$ when $x = 4$.

If $y$ is proportional to $x^n$ and doubling $x$ causes $y$ to increase by a factor of 8, find $n$.

The variable $y$ varies inversely as the square root of $x$ with proportionality constant 15. Determine $y$ when $x=9$.

For a certain process, $y$ varies as the inverse fifth power of $x$ with constant $k = 32$. Find $x$ when $y = 1$.

If $y$ is inversely proportional to the fourth power of $x$ with $k=9$, find the positive value of $x$ when $y=10$.