Question

If the radius of a planet is larger than that of Farth by a factor of 5.8 , how much bigger is the volume of the planet than Earth's?

Short answer

Answer: The volume of the planet is approximately 195.112 times larger than Earth's volume.

Step-by-step solution

Write down the formula for the volume of a sphere

The formula for the volume of a sphere is given by \(V = \frac{4}{3}\pi r^3\), where V is the volume and r is the radius of the sphere.

Set up the equation for the planet's radius and Earth's radius

Let's denote Earth's radius as r_E and the planet's radius as r_P. According to the problem, the planet's radius is 5.8 times larger than Earth's radius, so we can write: \[r_P = 5.8 \times r_E\]

Calculate the volume of Earth and the planet

Using the formula for the volume of a sphere, we can write the volume of Earth \(V_E\) and the volume of the planet \(V_P\) as: \[V_E = \frac{4}{3}\pi {r_E^3}\] \[V_P = \frac{4}{3}\pi {r_P^3}\]

Substitute the expression for the planet's radius from step 2 into the equation for the planet's volume

Now we can substitute \(r_P = 5.8 \times r_E\) into the equation for \(V_P\): \[V_P = \frac{4}{3}\pi (5.8 \times r_E)^3\]

Calculate the ratio of the volumes of the planet and Earth

To find how much bigger the planet's volume is than Earth's, we need to calculate the ratio \(R\) of their volumes: \[R = \frac{V_P}{V_E}\] Substitute the expressions for the volumes of the planet and Earth: \[R = \frac{\frac{4}{3}\pi (5.8 \times r_E)^3}{\frac{4}{3}\pi {r_E^3}}\]

Simplify the ratio and calculate the result

To get the ratio, simplify the equation: \[R = \frac{(5.8 \times r_E)^3}{r_E^3}\] Now, cancel out the \(r_E^3\) in the numerator and the denominator: \[R = (5.8)^3\] Calculate the cube of 5.8: \[R = 195.112\]

Indicate the conclusion

The volume of the planet is approximately 195.112 times larger than Earth's volume.

Key concepts

Volume of a Sphere

Understanding the volume of a sphere is essential when comparing sizes of different spherical objects, such as planets. The volume is the amount of space that a sphere occupies, and its calculation is based on the sphere's radius. The radius is the distance from the center point of the sphere straight to its surface. The formula used to calculate a sphere's volume is

\[\begin{equation}V = \frac{4}{3}\pi r^3\end{equation}\]
where \(V\) represents volume and \(r\) is the radius of the sphere. The \(\pi\) in the formula is a constant approximately equal to 3.14159, representing the ratio of a circle's circumference to its diameter. The cube of the radius, denoted as \(r^3\), amplifies the effect of the radius on the volume significantly, which is an important concept when comparing volumes of spheres with different radii.

Scaling Factor in Volume Calculation

In volume calculations, the scaling factor plays a critical role. A scaling factor is a multiplier that dilates or contracts dimensions proportionally. When comparing geometric figures, if one dimension of an object is scaled by a certain factor, the volume scales by the cube of that factor, due to dimensions being three-dimensional.

For spheres, if the radius of one sphere is scaled by a factor \(s\), the new volume \(V')\) will be related to the original volume \(V)\) by the equation:
\[\begin{equation}V' = s^3 \times V\end{equation}\]
This relationship is derived from the fact that the volume of a sphere formula contains the radius cubed. If you increase the radius of a sphere, every linear dimension of the sphere increases by the scaling factor, but since volume is a measure of three-dimensional space, the scaling effect is seen to the power of three, making the volume change dramatically with even small changes in the radius.

Volume Comparison Between Spheres

Comparing volumes of spheres involves understanding the comparative sizes of the spheres in question. The volume ratio of two spheres reflects how many times bigger or smaller one sphere is relative to the other.

\[\begin{equation}R = \frac{V_{\text{sphere A}}}{V_{\text{sphere B}}}\end{equation}\]
where \(R\) represents the ratio of the volumes, and \(V_{\text{sphere A}}\) and \(V_{\text{sphere B}}\) represent the volumes of sphere A and sphere B, respectively. To calculate this ratio, you need the radii of both spheres. Following the formula, if the radius of one sphere is a multiple of the other – for example, if one radius is \(5.8\) times larger – the volume ratio can be calculated by cubing the scaling factor (5.8 in this case). Thus, for spheres of different sizes, this volume ratio can be astonishingly large, despite seemingly modest differences in radius, highlighting the importance of accounting for the three-dimensional nature of these geometric bodies when comparing their sizes.