The population of a certain species of bacteria, $N$, is inversely proportional to the square root of the temperature, $t^\circ\mathrm{C}$, where $t>0$. Find the percentage decrease in the temperature when the population is doubled. Answer ... %

1. Write the inverse proportional relationship between population and temperature.

Since $N$ is inversely proportional to $\sqrt{t}$,
$N=\frac{k}{\sqrt{t}}$
where $k$ is a constant.

2. Compare the original and new situations when the population is doubled.

Let the original temperature be $t$ and the new temperature be $T$.

Original population:
$N=\frac{k}{\sqrt{t}}$
New population is doubled:
$2N=\frac{k}{\sqrt{T}}$

3. Find the new temperature as a fraction of the original temperature.

Using $N=\dfrac{k}{\sqrt{t}}$,
$2\left(\frac{k}{\sqrt{t}}\right)=\frac{k}{\sqrt{T}}$
Cancel $k$:
$\frac{2}{\sqrt{t}}=\frac{1}{\sqrt{T}}$
Hence
$\sqrt{t}=2\sqrt{T}$
Square both sides:
$t=4T$
$T=\frac{t}{4}$

4. Calculate the percentage decrease in temperature.

Decrease in temperature:
$t-\frac{t}{4}=\frac{3t}{4}$
Percentage decrease:
$\frac{\frac{3t}{4}}{t}\times 100\%=75\%$