$A$ body is heated to $110^{\circ} C$ and placed in air at $10^{\circ} C$. After $1$ hour,its temperature is $60^{\circ} C$. The additional time required for it to cool to $30^{\circ} C$ is

Let $\Gamma$ denote a curve $y = y(x)$ which is in the first quadrant and let the point $(1,0)$ lie on it. Let the tangent to $\Gamma$ at a point $P$ intersect the $y$-axis at $Y_p$. If $PY_p$ has length $1$ for each point $P$ on $\Gamma$,then which of the following options is/are correct?
$(1)$ $y=\ln\left(\frac{1+\sqrt{1-x^2}}{x}\right)-\sqrt{1-x^2}$
$(2)$ $xy^{\prime}+\sqrt{1-x^2}=0$
$(3)$ $y=-\ln\left(\frac{1+\sqrt{1-x^2}}{x}\right)+\sqrt{1-x^2}$
$(4)$ $xy^{\prime}-\sqrt{1-x^2}=0$

The population of a town increases at a rate proportional to the population at that time. If the population increases from $40,000$ to $80,000$ in $40$ years,then the population in another $40$ years will be (in $,000$)

$A$ spherical metal ball at $80^{\circ} C$ cools in $5 \text{ minutes}$ to $60^{\circ} C$ in a surrounding temperature of $20^{\circ} C$. The temperature of the ball after $20 \text{ minutes}$ is approximately: (in $^{\circ} C$)

The population $P(t)$ of a certain mouse species at time $t$ satisfies the differential equation $\frac{dP(t)}{dt} = 0.5 P(t) - 450$. If $P(0) = 850$,then the time at which the population becomes zero is

If a body is heated to $110^{\circ} C$ and placed in air at $10^{\circ} C$,and after $1 \text{ hour}$ its temperature is $60^{\circ} C$,then the additional time required for it to cool to $30^{\circ} C$ is