Question

For the planet Mars, calculate the distance around the Equator, the surface area, and the volume. The radius of Mars is \(3.39 \cdot 10^{6} \mathrm{~m}\)

Short answer

Question: Calculate the circumference, surface area, and volume of the planet Mars, given its radius is 3.39 x 10^6 meters.

Step-by-step solution

Calculate the circumference

Using the formula for the circumference, \(C = 2 \times \pi \times r\), we can plug in the value for the radius of Mars, which is \(3.39 \cdot 10^6 \mathrm{~m}\). $$ C = 2 \times \pi \times (3.39 \cdot 10^6 \mathrm{~m}) $$ Now, you can calculate the value of \(C\) to find the circumference.

Calculate the surface area

Using the formula for the surface area, \(A = 4 \times \pi \times r^2\), we can plug in the value for the radius of Mars. $$ A = 4 \times \pi \times (3.39 \cdot 10^6 \mathrm{~m})^2 $$ Now, calculate the value of \(A\) to find the surface area.

Calculate the volume

Using the formula for the volume of a sphere, \(V = \frac{4}{3} \times \pi \times r^3\), we can plug in the value for the radius of Mars. $$ V = \frac{4}{3} \times \pi \times (3.39 \cdot 10^6 \mathrm{~m})^3 $$ Now, calculate the value of \(V\) to find the volume. With the values of circumference, surface area, and volume calculated, you now have the information requested for the planet Mars.

Key concepts

Understanding the Circumference of Mars

Grasping the concept of the circumference of a planet like Mars is not just about doing calculations; it embodies understanding the scale and the composition of the planet itself. The circumference is fundamentally the distance all the way around the equator of Mars. To calculate the circumference, the formula employed is simply the basic circumference of a circle, which is \( C = 2 \times \bold\temp\pi \times r \bold\temp\).

Using Mars' radius of \(3.39 \times 10^6 \text{ m} \bold\temp\), this formula reveals the planet's vast size. By calculating the circumference, students get a tangible feel for the actual size of Mars compared to Earth and can start to conceptualize the distances involved in space travel.

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• How does Mars' circumference compare to Earth's?
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• What implications does the planet's circumference have for phenomena like a day's length on Mars?
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• Why is understanding the circumference critical for missions to Mars?

These are the kinds of questions that extend the learning beyond mere numbers and engage students with the practicalities and wonders of space exploration.

Calculating the Surface Area of Mars

Diving into the concept of surface area, it's essential to acknowledge its importance in various scientific calculations, including understanding the climate and physical landscape of a planet. For Mars, calculating the surface area involves using the formula \( A = 4 \times \bold\temp\pi \times r^2 \bold\temp\), where \( r \bold\temp\) is the radius of Mars.

The resulting calculation of the surface area allows us to contemplate the expanse of Mars' terrain that's covered with craters, volcanoes, and vast empty deserts. It offers a window into planetology and the ability to compare it with Earth’s diverse habitats.

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• Surface area calculation helps in understanding the potential for human colonization.
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• It plays a role in the analysis of Mars' atmosphere.
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• What does the surface area tell us about Mars’ capacity to harbor water or life?

Envisioning a planet's surface area goes beyond the digits we calculate; it is about interpreting and realizing the relationship between area and planetary features.

Understanding the Volume of Mars

Lastly, when we look at volume, we're venturing into a three-dimensional space that defines the quantity inside the spherical shape of Mars. The formula used here is \( V = \frac{4}{3} \times \bold\temp\pi \times r^3 \bold\temp\), which provides us with a measure of how much space Mars occupies.

The volume of Mars tells us not only about its size but also about its mass and density when paired with other data, which are fundamental to understanding the planet's gravitational pull and its ability to retain an atmosphere. The concept of volume is significant in multiple areas such as:

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• Gravitational studies and how Mars interacts with its moons and the Sun.
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• The study of Mars' core and interior structure.
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• How volume relates to the planet’s formation and evolutionary history.

By comprehending the volume, students are encouraged to think about planetary formation and characteristics that make each planet unique in our solar system. It's a step towards unraveling the mysteries that the red planet holds.