a) The density of the planet Saturn is \(\boxed{0.619\,{{\text{g}} \mathord{\left/ {\vphantom {{\text{g}} {{\text{c}}{{\text{m}}^{\text{3}}}}}} \right. \kern-
ulldelimiterspace} {{\text{c}}{{\text{m}}^{\text{3}}}}}}\).
b) The surface area of the planet Saturn is \(\boxed{4.57 \times {{10}^{16}}\,{{\text{m}}^{\text{2}}}}\).
Further Explanation:
Given:
The radius of the planet Saturn is \(6.03 \times {10^7}\,{\text{m}}\).
The mass of the planet Saturn is \(5.68 \times {10^{26}}\,{\text{kg}}\).
Concept:
The planet is considered to be a huge spherical body. Therefore, the volume of the planet is expressed as:
\(V = \frac{4}{3}\pi {r^3}\)
Substitute the value of \(r\) in the above expression.
\(\begin{aligned}V&=\frac{4}{3}\pi\times{\left({6.03\times{{10}^9}\,{\text{cm}}}\right)^3}\\&=4.189\times2.19\times{10^{29}}\\&=9.17\times{10^{29}}\,{\text{c}}{{\text{m}}^{\text{3}}}\\\end{aligned}\)
Part (a):
The density of the planet can be expressed as:
\(\rho = \frac{{{\text{mass}}}}{{{\text{Volume}}}}\)
Substitute the values of mass and volume of the planet.
\(\begin{aligned}\rho&=\frac{{5.68\times{{10}^{26}}\,{\text{kg}}}}{{9.17 \times{{10}^{29}}\,{\text{c}}{{\text{m}}^{\text{3}}}}}\\&=\frac{{5.68\times{{10}^{29}}\,{\text{g}}}}{{9.17\times{{10}^{29}}\,{\text{c}}{{\text{m}}^{\text{3}}}}}\\&=0.619\,{{\text{g}}\mathord{\left/{\vphantom{{\text{g}}{{\text{c}}{{\text{m}}^{\text{3}}}}}}\right. \kern-
ulldelimiterspace}{{\text{c}}{{\text{m}}^{\text{3}}}}}\\\end{aligned}\)
Thus, the density of the planet Saturn is \(\boxed{0.619\,{{\text{g}} \mathord{\left/ {\vphantom {{\text{g}} {{\text{c}}{{\text{m}}^{\text{3}}}}}} \right. \kern-
ulldelimiterspace} {{\text{c}}{{\text{m}}^{\text{3}}}}}}\).
Part (b):
The total surface area of the spherical planet is given as:
\(A = 4\pi {r^2}\)
Substitute the value of \(r\) in the above expression.
\(\begin{aligned}A&=4\pi\times{\left({6.03\times{{10}^7}\,{\text{m}}}\right)^2}\\&= 12.56 \times3.63\times{10^{15}}\\&=4.569\times{10^{16}}\,{{\text{m}}^{\text{2}}} \\ \end{aligned}\)
Thus, the surface area of the planet Saturn is \(\boxed{4.57 \times {{10}^{16}}\,{{\text{m}}^{\text{2}}}}\).
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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.
