Answer :
Answer:
[tex]2.07\times10^{26}kg[/tex]
Explanation:
Mass of the planet can be found by integrating the density and the volume.
The volume of the planet would be:
dV = 4πr²dr
dM= ρ(r) 4πr²dr
Total mass:
[tex]\int_{0}^{M} dm=\int_{0}^{R_o} \rho(r)4\pi r^2dr\\ \Rightarrow \int_{0}^{M} dm=\int_{0}^{R_o} \rho_o(1-\frac{\alpha r}{R_o})4\pi r^2dr\\ \Rightarrow \int_{0}^{M} dm=4\pi \rho_o \int_{0}^{R_o}(r^2-\frac{\alpha r^3}{R_o})dr\\ \Rightarrow M= 4\pi (\rho_o\frac{r^3}{3}-\frac{\alpha r^4}{4R_o})[/tex]
Insert the limits and then substitute the values:
[tex]M=4\pi \rho_o (\frac{R_o^3}{3}-\frac{\alpha R_o^3}{4})\\ M =4\pi \rho_oR_o^3 (\frac{1}{3}-\frac{\alpha}{4})\\ M= 4\times 3.14 \times 3800.0\times (25.12\times10^6)^3(\frac{1}{3}-\frac{0.24}{4})\\ M=2.07\times10^{26}kg[/tex]