Q$4.$ Particle size function

The measurement of fog in an airport shows that the droplet size distribution of the fog may be represented by the equations:

$50x15m,N_0(x)=0\mathrm{.}83+0\mathrm{.}043ln(x)$

$15x4m,N_0(x)=0\mathrm{.}30+0\mathrm{.}239ln(x)$

$4x0\mathrm{.}5m,N_0(x)=0\mathrm{.}21+0\mathrm{.}304ln(x)$

$N_0(x)$ is the cumulative fraction by number.

$(1)$ Write down the questions for $n_0(x)$ for different particle size ranges.

$(2)$ What is the mean droplet size by number?

$(3)$ What is the mean droplet size by mass?

$(4)$ The specific surface area per unit volume $(S_v$ of the fog is the total surface area divided by the total volume, in number distribution terms $($assuming spherical droplets$).$ What is the $S_v$ for the fog?

$(5)$ What is the Sauter mean diameter of droplet in the fog?