In a certain culture of bacteria,the rate of increase is proportional to the number present. If there are $10^4$ at the end of $3$ hours and $4 \cdot 10^4$ at the end of $5$ hours,then there were $\qquad$ in the beginning.

$A$ population grows at the rate of $10 \%$ of the population per year. How long does it take for the population to double?

The rate of increase of the population of a city is proportional to the population present at that instant. In the period of $40$ years,the population increased from $30,000$ to $40,000$. At any time $t$,the population is given by $P(t) = (a)(b)^{\frac{t}{40}}$. Then the values of $a$ and $b$ are respectively:

Water at $100^{\circ} C$ cools in $15 \text{ minutes}$ to $75^{\circ} C$ in a room temperature of $25^{\circ} C$. Then the temperature of water after $30 \text{ minutes}$ is

The number of differentiable functions $y: (-\infty, \infty) \rightarrow [0, \infty)$ satisfying $y^{\prime} = 2 \sqrt{y}$ and $y(0) = 0$ is

The family of curves in which the sub-tangent at any point to any curve is double the abscissa is given by