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11 January, 01:53

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.

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  1. 11 January, 02:05
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    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.
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