/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 39 The surface area of a sphere is ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 39

The surface area of a sphere is proportional to the square of its radius. The radius of the Moon is only about one-quarter that of Earth. How does the surface area of the Moon compare with that of Earth?

Short Answer

Expert verified
The surface area of the Moon is \(\frac{1}{16}\) of the surface area of Earth.

Step by step solution

01

- Understand the relationship

The surface area of a sphere is given by the formula \[ A = 4\pi r^2 \], where \(r\) is the radius. From this equation, we know that the surface area is proportional to the square of the radius.
02

- Introduce the ratio of radii

Given that the radius of the Moon is one-quarter that of the Earth, let the radius of the Earth be \(r_E\). Then, the radius of the Moon \(r_M\) is \[ r_M = \frac{1}{4} r_E \].
03

- Apply the ratio to the surface area

According to the formula for surface area, we have: \[ A_E = 4\pi r_E^2 \] for Earth and \[ A_M = 4\pi \left(\frac{1}{4} r_E\right)^2 \] for the Moon.
04

- Simplify the Moon's surface area formula

Simplify the surface area formula for the Moon: \[ A_M = 4\pi \left(\frac{1}{4} r_E\right)^2 = 4\pi \left(\frac{1}{16} r_E^2\right) = \frac{1}{4} \pi r_E^2 \].
05

- Compare the surface areas

To compare the surface areas, look at the ratio \[ \frac{A_M}{A_E} = \frac{\frac{1}{4} \pi r_E^2}{4\pi r_E^2} = \frac{1}{16} \]. Therefore, the surface area of the Moon is \(\frac{1}{16}\) of the surface area of Earth.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

sphere surface area
In geometry, a sphere is a perfectly round three-dimensional shape. One important characteristic of a sphere is its surface area, which is the total area that the surface of the sphere covers.
To find the surface area of a sphere, we use the formula: \(\text{A} = 4\text{蟺}\text{r}^2\), where \(r\) is the radius of the sphere.
This formula essentially tells us how much space the surface of the sphere would cover if it were laid out flat.
Understanding this concept is essential because it forms the basis for comparing the surface areas of different spheres, such as the Earth and the Moon.
radius to surface area relationship
The surface area of a sphere is closely related to its radius. The formula \(A = 4\text{蟺}r^2\) demonstrates that the surface area is proportional to the square of the radius.
This means if you double the radius of a sphere, its surface area doesn鈥檛 just double; it actually increases by a factor of four (since \(2^2 = 4\)).
Therefore, even a small change in the radius of a sphere can lead to a significant change in its surface area.
In our comparison of the Moon and Earth, the radius of the Moon is one-quarter of the Earth's radius. Consequently, the surface area of the Moon will be considerably smaller.
mathematical ratios
Mathematical ratios help us understand the relationship between different quantities.
In our exercise, we used the ratio of the radii of the Moon and Earth to find the ratio of their surface areas.
Given that the Moon's radius is one-quarter that of the Earth, we set up the equation \( r_M = \frac{1}{4} r_E \).
Applying this to the surface area formula, we found that the Moon's surface area \( A_M = 4\text{蟺}\frac{1}{16}r_E^2 \).
By comparing it to the Earth's surface area \(A_E = 4\text{蟺}r_E^2 \), we determined that the Moon鈥檚 surface area is \(\frac{1}{16}\) of Earth's surface area.
This illustrates how ratios can simplify complex comparisons.
Moon vs Earth
Now let's dive into how the surface areas of the Moon and Earth compare.
Using the relationship between their radii and surface areas, we established that the Moon鈥檚 radius is one-quarter that of Earth's.
This smaller radius leads to an even smaller surface area. In fact, the Moon's surface area is just \( \frac{1}{16} \) of Earth's surface area.
To put it simply, if you think of the Earth as a large sphere, the Moon is a much smaller sphere with significantly less surface area.
This comparison is fascinating and highlights the vast difference between the two celestial bodies, despite them both being spheres.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

(a) If it takes about 8 minutes for light to travel from the Sun to Earth, and Neptune is 30 times farther from Earth than the Sun is, how long does it take light to reach Earth from Neptune? (b) Radio waves travel at the speed of light. What does this fact imply about the problems you would have if you tried to conduct a two-way conversation between Earth and a spacecraft orbiting Neptune?

New York is 2,444 miles from Los Angeles. What is that distance in car-hours? In car-days? (Assume a travel speed of \(70 \mathrm{mph} .)\)

"We are stardust" means that a. Earth exists because of the collision of two stars. b. the atoms in our bodies have passed through (and in many cases formed in stars. c. Earth is primarily formed of material that used to be in the Sun. d. Earth and the other planets will eventually form a star.

Throughout this book, we will examine how discoveries in astronomy and space are covered in the media. Go to your favorite news website (or to one assigned by your instructor) and find a recent article about astronomy or space. Does this website have a separate section for science? Is the article you selected based on a press release, on interviews with scientists, or on an article in a scientific journal? Use Google News or the equivalent to see how widespread the coverage of this story is. Have many newspapers carried it? Has it been picked up internationally? Has it been discussed in blogs? Do you think this story was interesting enough to be covered?

Imagine yourself living on a planet orbiting a star in a very distant galaxy. What does the cosmological principle tell you about the physical laws at this distant location?
See all solutions

Recommended explanations on Physics Textbooks

Atoms and Radioactivity

Read Explanation

Space Physics

Read Explanation

Magnetism and Electromagnetic Induction

Read Explanation

Translational Dynamics

Read Explanation

Geometrical and Physical Optics

Read Explanation

Torque and Rotational Motion

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.