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24 May, 09:24

The mean diameters of planets A and B are 8.1 * 103 km and 1.4 * 104 km, respectively. The ratio of the mass of planet A to that of planet B is 0.96. (a) What is the ratio of the mean density of A to that of B

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  1. 24 May, 09:38
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    dA/dB = 4.955

    Approximately, the ratio is 5/1

    (Where dA is mean density for planet A while dB is mean density for planet B)

    Explanation:

    Mass of A = mA

    Mass of B = mB

    mA/mB = 0.96

    Mean radius for A = mA = (8.1 * 10^3) / 2 = 4.05 * 10^3 km

    Mean radius for B = mB = (1.4 * 10^4) / 2

    = 7*10^3km

    Density = mass/volume

    Volume of a sphere = 4/3Πr3

    Mean volume for A = (4/3) * Π * (4.05 * 10^3) ^3

    = 2.784 * 10^11 km3

    Mean volume for B = 4/3*Π * (7*10^3) ^3

    = 1.437 * 10^12km3

    Since m/v = d (where m = mass, v = volume and d = density)

    mA = 2.784 * 10^11 km3 * dA ... equation 1

    mB = 1.437 * 10^12km3 * dB ... equation 2

    but mA/mB = 0.96

    mA = 0.96 * mB

    substitute for mA in equation 1

    0.96 * mB = 2.784 * 10^11 x dA equation 3

    Substitute for mB in equation 3 ...

    (refer to equation 2)

    0.96*1.437*10^12 * dB = 2.784 * 10^11 * dA ... equation 4

    divide through by the coefficient of dA

    dA = (0.96*1.437*10^12*dB) / (2.784 * 10^11)

    divide through by dB

    dA/dB = 4.955

    therefore, the ratio of dA to dB is 5/1

    Therefore, the mean density of A is almost five times that of B
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