Respuesta :
You wouldn't know it from reading the question,
but the volume of a sphere is
4/3 π (radius)³ .
So the volume of Saturn is
(4/3 π) (6.03 x 10⁷ m)³ = about 9.184 x 10²³ m³
and its density is (mass) / (volume)
= (5.68 x 10²⁶ kg) / (9.184 x 10²³ m³)
Sadly, we need this to be in units of ' gram/cm³ ' so we need to
account for some conversion of units.
= (5.68 x 10²⁶ kg) · (1,000 g/kg) / (9.184 x 10²³ m³) · (10⁶ cm³/m³)
= (5.68 x 10²⁹ grams) / (9.184 x 10²⁹ cm³)
= 0.618 gram/cm³ .
Look at that !
The density of the planet Saturn is less than ' 1 '.
If you had a big enough bathtub full of water, Saturn would float in it !
The surface area of a sphere is
4 π (radius)²
= (4 π) (6.03 x 10⁷ m)²
= (4 π) (36.4 x 10¹⁴ m²)
= 4.57 x 10¹⁶ m²
a) The density of the planet Saturn is [tex]\boxed{0.619\,{{\text{g}} \mathord{\left/ {\vphantom {{\text{g}} {{\text{c}}{{\text{m}}^{\text{3}}}}}} \right. \kern-\nulldelimiterspace} {{\text{c}}{{\text{m}}^{\text{3}}}}}}[/tex].
b) The surface area of the planet Saturn is [tex]\boxed{4.57 \times {{10}^{16}}\,{{\text{m}}^{\text{2}}}}[/tex].
Further Explanation:
Given:
The radius of the planet Saturn is [tex]6.03 \times {10^7}\,{\text{m}}[/tex].
The mass of the planet Saturn is [tex]5.68 \times {10^{26}}\,{\text{kg}}[/tex].
Concept:
The planet is considered to be a huge spherical body. Therefore, the volume of the planet is expressed as:
[tex]V = \frac{4}{3}\pi {r^3}[/tex]
Substitute the value of [tex]r[/tex] in the above expression.
[tex]\begin{aligned}V&=\frac{4}{3}\pi\times{\left({6.03\times{{10}^9}\,{\text{cm}}}\right)^3}\\&=4.189\times2.19\times{10^{29}}\\&=9.17\times{10^{29}}\,{\text{c}}{{\text{m}}^{\text{3}}}\\\end{aligned}[/tex]
Part (a):
The density of the planet can be expressed as:
[tex]\rho = \frac{{{\text{mass}}}}{{{\text{Volume}}}}[/tex]
Substitute the values of mass and volume of the planet.
[tex]\begin{aligned}\rho&=\frac{{5.68\times{{10}^{26}}\,{\text{kg}}}}{{9.17 \times{{10}^{29}}\,{\text{c}}{{\text{m}}^{\text{3}}}}}\\&=\frac{{5.68\times{{10}^{29}}\,{\text{g}}}}{{9.17\times{{10}^{29}}\,{\text{c}}{{\text{m}}^{\text{3}}}}}\\&=0.619\,{{\text{g}}\mathord{\left/{\vphantom{{\text{g}}{{\text{c}}{{\text{m}}^{\text{3}}}}}}\right. \kern-\nulldelimiterspace}{{\text{c}}{{\text{m}}^{\text{3}}}}}\\\end{aligned}[/tex]
Thus, the density of the planet Saturn is [tex]\boxed{0.619\,{{\text{g}} \mathord{\left/ {\vphantom {{\text{g}} {{\text{c}}{{\text{m}}^{\text{3}}}}}} \right. \kern-\nulldelimiterspace} {{\text{c}}{{\text{m}}^{\text{3}}}}}}[/tex].
Part (b):
The total surface area of the spherical planet is given as:
[tex]A = 4\pi {r^2}[/tex]
Substitute the value of [tex]r[/tex] in the above expression.
[tex]\begin{aligned}A&=4\pi\times{\left({6.03\times{{10}^7}\,{\text{m}}}\right)^2}\\&= 12.56 \times3.63\times{10^{15}}\\&=4.569\times{10^{16}}\,{{\text{m}}^{\text{2}}} \\ \end{aligned}[/tex]
Thus, the surface area of the planet Saturn is [tex]\boxed{4.57 \times {{10}^{16}}\,{{\text{m}}^{\text{2}}}}[/tex].
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Answer Details:
Grade: High School
Subject: Physics
Chapter: Units and Measurement
Keywords:
Radius of the planet, Saturn, mass of Saturn, density, grams per cubic centimeter, 5.68x10^26 kg, 6.03x10^7 m, volume of sphere.
