What is the total surface area of these N spherical droplet?

Question

The complete questions is The basic function of a carburetor of an automobile is to atomize the gasoline and mix it with air to promote rapid combustion. As an example, assume that 30 cm^3 of gasoline is atomized into N spherical droplets, each with a radius of 2.0 x 10^-5 m. What is the total surface area of these N spherical droplet? Solution with explanation please. TIA!

Answer 1

Radius of droplet=2×10⁻⁵m=0.002cm.

The volume of a droplet is (4/3)(0.002)³π cc. So (4N/3)(0.002)³π=30.

From this N=(45/2)/(0.002)³π)=2.8125×10⁹/π.

The surface area of each droplet is 4π(0.002)²=1.6π×10⁻⁵ sq cm.

Total surface area is 2.8125×10⁹/π × 1.6π×10⁻⁵ sq cm = 4.5×10⁴ sq cm = 45000 sq cm.

(If r=radius of droplet, (4/3)πr³ is its volume. If V is the volume of the gasoline, N=V/((4/3)πr³)=3V/(4πr³). Surface area of droplet=4πr². Total surface area=4πr² × 3V/(4πr³)=3V/r=90/0.002=45000 sq cm. Using algebra first greatly simplifies the calculation.)