Respuesta :
Answer:
The total mass of this planet is [tex]2.0689\times10^{26}\ kg[/tex]
Explanation:
Given that,
Radius [tex]R_{0}=25.12\times10^{6}\ m[/tex]
Density [tex]\rho_{0}=3800.0\ kg/m^3[/tex]
Central density[tex]\alpha=0.24[/tex]
The density of a certain planet varies with radial distance as
[tex]\rho(r)=\rho_{0}(1-\dfrac{\alpha r}{R_{0}})[/tex]
We need to calculate the total mass of this planet
Using formula of density
[tex]\rho=\dfrac{M}{V}[/tex]
[tex]M=\rho\times V[/tex]
On integrating
[tex]M=\int_{0}^{R_{0}}{\rho(r)\times4\pi r^2 dr}[/tex]
Put the value of [tex]\rho{r}[/tex] into the formula
[tex]M=\int_{0}^{R_{0}}{\rho_{0}(1-\dfrac{\alpha r}{R_{0}})\times4\pi r^2 dr}[/tex]
[tex]M=\rho_{0}\times 4\pi\int_{0}^{R_{0}}{(r^2-\dfrac{\alpha r^3}{R_{0}})dr}[/tex]
[tex]M=\rho_{0}\times 4\pi\times(\dfrac{r^3}{3}-\dfrac{\alpha\times r^4}{4\times R_{0}})_{0}^{R_{0}}[/tex]
[tex]M=4\pi\times\rho_{0}\times R_{0}^3(\dfrac{4-3\alpha}{12})[/tex]
Put the value into the formula
[tex]M=4\pi\times3800.0\times(25.12\times10^{6})^3(\dfrac{4-3\times0.24}{12})[/tex]
[tex]M=2.0689\times10^{26}\ kg[/tex]
Hence, The total mass of this planet is [tex]2.0689\times10^{26}\ kg[/tex]
