Question

If the radius of a planet is larger than that of Earth 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 the volume of Earth.

Step-by-step solution

Write the formula for the volume of a sphere

The formula for the volume of a sphere is given by V = (4/3)πr^3, where V is the volume and r is the radius of the sphere.

Find the ratio of the radii of the planet and Earth

We are given that the radius of the planet is 5.8 times larger than the radius of Earth. Let r_E denote the radius of Earth, and r_P denote the radius of the planet. Then, r_P = 5.8 * r_E.

Substitute the ratio of the radii into the volume formula

Now, we need to find the ratio of the volumes of the planet and Earth. Let V_E = (4/3)π(r_E^3) be the volume of Earth and V_P = (4/3)π(r_P^3) be the volume of the planet. We can find the ratio of V_P to V_E using the formula: V_ratio = V_P / V_E. We have V_ratio = ( (4/3)π(r_P^3) ) / ( (4/3)π(r_E^3) ), substituting r_P = 5.8 * r_E, we get: V_ratio = ( (4/3)π((5.8 * r_E)^3) ) / ( (4/3)π(r_E^3) )

Simplify the expression for V_ratio

Now, we need to simplify the expression for V_ratio: V_ratio = ( (4/3)π(5.8^3 * r_E^3) ) / ( (4/3)π(r_E^3) ) Notice that (4/3)π and r_E^3 can be canceled from both numerator and denominator: V_ratio = 5.8^3

Calculate the value of V_ratio

Finally, we need to calculate the value of V_ratio: V_ratio = (5.8)^3 = 195.112 So, the volume of the planet is approximately 195.112 times larger than the volume of Earth.

Key concepts

Volume of a Sphere

Understanding the volume of a sphere is foundational when comparing celestial bodies like planets. The volume, which describes the space that a three-dimensional object occupies, is particularly simple to compute for spheres because of their symmetric properties. For any sphere, the volume is related to the cube of its radius and can be found using the formula:

\[ V = \frac{4}{3}\pi r^3 \]
In this expression, \( V \) is the volume of the sphere, \( r \) is the radius, and \( \pi \) is the constant Pi, approximately equal to 3.14159. This equation suggests that the volume of a sphere increases drastically with a small increase in the radius, due to the radius being raised to the power of three.

Radius Factor

In many textbook problems, like the one concerning planetary volumes, you'll encounter the term 'radius factor'. This refers to how many times larger or smaller one radius is compared to another. In our exercise, the radius of a planet is said to be larger by a factor of 5.8 compared to Earth. This is represented mathematically by:

\[ r_P = 5.8 \cdot r_E \]
where \( r_P \) is the radius of the planet, and \( r_E \) is the radius of Earth. This factor is crucial because it plays a pivotal role in calculating the volume difference due to the cubed relationship between the radius and volume of a sphere. As the radius changes, the volume changes to the power of three of the radius factor, leading to a significant amplification even with relatively small factors.

Volume Comparison

Comparing volumes, especially when dealing with planets or spheres, involves utilizing the radius and the volume formula. Because the formula for the volume contains the radius raised to the third power, small changes in the radius result in amplified changes in volume. To compare the volume of two spheres, we calculate the ratio of their volumes by dividing one by the other. Taking the exercise as an example:

\[ V_{\text{ratio}} = \left( \frac{r_P}{r_E} \right)^3 \]
By substituting the radius factor into this formula, we isolate how much larger one volume is relative to another. In simpler terms, if a planet's radius is 5.8 times Earth's, the planet's volume is \(5.8^3\), or about 195.112 times greater than Earth’s volume. This volume comparison emphasizes the exponential nature of volumetric growth in relation to radius size, which is a fundamental concept in not only geometry but also in understanding the scales of physical objects in our universe.

Visualizing Volume Changes

Imagine inflating a balloon. The radius might double or triple, but the amount of space it occupies (the volume) grows by the cube of the change in radius. So, a seemingly modest increase in radius produces a much larger balloon than you might intuitively expect, and the same principle applies to planets and their volumes.