Recently, some information about a distant planet was learned. It has a radius of 6000000 meters, and the density of the atmosphere as a function of the height h (in meters) above the surface of the planet is given by δ (h) = 3h+6000000 kilograms per cubic meter. Calculate the mass of the portion of the atmosphere from h=0 to h=57.

M = 9.8 * 10²²kg

Explanation:

This question involves the mass, density and volume relationship. The density of the atmosphere varies with the height above the surface of the planet. Given the density function δ (h) = 3h+6000000. We can then calculate the value of the densities at the two given altitudes (height above the planet surface).

We will use the relationship between the mass, volume and density.

M = ρ * V

At h = 0m (the surface of the planet)

The radius of the planet = 6*10⁶m

ρ = 3h+6000000 = 3*0 + 6000000

= 6000000 = 6*10⁶ kg/m³

V = 4/3 * r³ (volume of a sphere)

= 4/3 * (6*10⁶) ³ = 2.88 * 10²⁰ m³

M = 6*10⁶ * 2.88 * 10²⁰ = 1.728 * 10²⁷

At a height of h = 57m

r = 6000000 + 57 = 6000057m

V = 4/3 * (6000057) ³ = 2.880082*10²⁰

ρ = 3h+6000000 = 3*57 + 6000000

= 6000171 kg/m³

M = 6000171 * 2.880082 * 10²⁰

= 1.728098 * 10²⁷

The mass of the atmosphere is the difference between the masses at the different altitudes.

So the mass of the atmosphere

= 1.728098 * 10²⁷ - 1.728000 * 10²⁷

= 9.8 * 10²² kg.