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

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