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Chapter 8: Problem 45

Assume that Earth and Mars are perfect spheres with radii of \(6,371 \mathrm{km}\) and \(3,390 \mathrm{km}\), respectively. a. Calculate the surface area of Earth. b. Calculate the surface area of Mars. c. If 0.72 (72 percent) of Earth's surface is covered with water, compare the amount of Earth's land area to the total surface area of Mars.

Short Answer

Expert verified
Earth's land area is about 142,818,052 km^2, Mars' surface area is about 144,371,391 km^2. The two areas are approximately equal.

Step by step solution

01

Calculate the Surface Area of Earth

Use the formula for the surface area of a sphere: \[A = 4\pi r^2\]. For Earth, the radius is 6,371 km. Plugging in this value: \[A_{Earth} = 4\pi (6,371)^2\]
02

Compute the Surface Area

Calculate the surface area of Earth, \[A_{Earth} = 4 * \pi * 6,371^2\] which approximately equals \[A_{Earth} \approx 4 * 3.14159 * 40,600,841 = 510,064,472 \text{ km}^2\]
03

Calculate the Surface Area of Mars

Using the same formula for the surface area of a sphere, with the radius of Mars being 3,390 km: \[A_{Mars} = 4\pi (3,390)^2\]
04

Compute the Surface Area

Calculate the surface area of Mars, \[A_{Mars} = 4 * \pi * 3,390^2\] which approximately equals \[A_{Mars} \approx 4 * 3.14159 * 11,492,100 = 144,371,391 \text{ km}^2\]
05

Calculate the Land Area of Earth

If 0.72 (72 percent) of Earth's surface is covered with water, the land area is 28 percent of the total surface area: \[\text{Land Area}_{Earth} = 0.28 * A_{Earth}\]. Using our value for Earth's surface area: \[\text{Land Area}_{Earth} = 0.28 * 510,064,472 \approx 142,818,052 \text{ km}^2\]
06

Compare Earth's Land Area to Mars' Surface Area

Now compare the land area of Earth to the total surface area of Mars: \[142,818,052 \text{ km}^2 (\text{Earth Land Area})\] vs. \[144,371,391 \text{ km}^2 (\text{Mars Surface Area})\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

surface area calculation
When you want to find the surface area of a sphere, you use a specific formula. This formula is: \(A = 4 \pi r^2\).
The letter \(A\) represents the surface area, while \(r\) is the radius of the sphere.
This formula works for any sphere, including planets like Earth and Mars.
You can think of the surface area as the total amount of space that covers the outside of the sphere.
To get the surface area, first square the radius (multiply the radius by itself), then multiply by \(\pi\) (which is approximately 3.14159), and finally, multiply by 4.
Let's figure out the surface area for Earth and Mars step-by-step to see how this works in practice.
Earth vs Mars
To compare Earth and Mars in terms of their surface areas, we can use the formula for the surface area of a sphere.
For Earth, the radius is 6,371 km. For Mars, the radius is 3,390 km.
Earth, being larger, has a significantly greater surface area compared to Mars.
After calculating using the sphere formula, we find:
\(A_{Earth} \approx 510,064,472 \text{ km}^2\)
\(A_{Mars} \approx 144,371,391 \text{ km}^2\)
So, Earth's surface area is a little over 3.5 times that of Mars.
sphere formula
Understanding the formula for the surface area of a sphere is crucial for these calculations.
The formula \(A = 4 \pi r^2\) comes from geometry.
It tells us that the surface area depends directly on the square of the radius.
In simpler terms, if you double the radius of a sphere, the surface area increases by a factor of four, because \(2^2 = 4\).
This relationship helps explain why Earth, with roughly twice the radius of Mars, has a much larger surface area.
Applying this formula to any spherical object will help you determine the surface area, as shown in the comparison between Earth and Mars.

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