3. Read the article "Planet". written by William A. Gutsch Jr. You will use information from this article to answer the following questions.

I'm sorry, but I cannot access specific external articles, including "Planet" by William A. Gutsch Jr. However, I can provide general information about the average distances of the planets from the Sun, represented in scientific notation, and help you with the calculations.

Here is a list of the average distances of the planets from the Sun in astronomical units (AU), where 1 AU is approximately \(1.496 \times 10^{11}\) meters. The distances of the planets in AU are approximately:

1. Mercury: \(0.39 , \text{AU} = 3.9 \times 10^{10} , \text{m}\)
2. Venus: \(0.72 , \text{AU} = 7.2 \times 10^{10} , \text{m}\)
3. Earth: \(1.00 , \text{AU} = 1.0 \times 10^{11} , \text{m}\)
4. Mars: \(1.52 , \text{AU} = 1.52 \times 10^{11} , \text{m}\)
5. Jupiter: \(5.20 , \text{AU} = 5.2 \times 10^{11} , \text{m}\)
6. Saturn: \(9.58 , \text{AU} = 9.58 \times 10^{11} , \text{m}\)
7. Uranus: \(19.22 , \text{AU} = 1.922 \times 10^{12} , \text{m}\)
8. Neptune: \(30.05 , \text{AU} = 3.005 \times 10^{12} , \text{m}\)

a. List of Distances from Each Planet to the Sun (in Scientific Notation):

1. Mercury: \(3.9 \times 10^{10} , \text{m}\)
2. Venus: \(7.2 \times 10^{10} , \text{m}\)
3. Earth: \(1.0 \times 10^{11} , \text{m}\)
4. Mars: \(1.52 \times 10^{11} , \text{m}\)
5. Jupiter: \(5.2 \times 10^{11} , \text{m}\)
6. Saturn: \(9.58 \times 10^{11} , \text{m}\)
7. Uranus: \(1.922 \times 10^{12} , \text{m}\)
8. Neptune: \(3.005 \times 10^{12} , \text{m}\)

b. Addition of Two Distances

Let’s add the distances of Earth and Mars.

\[ 1.0 \times 10^{11} + 1.52 \times 10^{11} = (1.0 + 1.52) \times 10^{11} = 2.52 \times 10^{11} , \text{m} \]

c. Subtraction of Two Distances

Let’s subtract the distance of Venus from the distance of Saturn.

\[ 9.58 \times 10^{11} - 7.2 \times 10^{10} = 9.58 \times 10^{11} - 0.72 \times 10^{11} = (9.58 - 0.72) \times 10^{11} = 8.86 \times 10^{11} , \text{m} \]

d. Multiplication of Two Distances

Let’s multiply the distances of Mercury and Venus.

\[ (3.9 \times 10^{10})(7.2 \times 10^{10}) = (3.9 \times 7.2) \times 10^{20} = 28.08 \times 10^{20} = 2.808 \times 10^{21} , \text{m}^2 \]

e. Division of Two Distances

Let’s divide the distance of Neptune by the distance of Jupiter.

\[ \frac{3.005 \times 10^{12}}{5.2 \times 10^{11}} = \frac{3.005}{5.2} \times 10^{12-11} = 0.577 \times 10^{1} = 5.77 , \text{AU} \]

These calculations illustrate how to manipulate scientific notation for the distances of planets from the Sun. If you need further details, feel free to ask!

Are you sure these are the correct distances?
the article is telling me something different.

I apologize for any discrepancy in the distances I provided. The average distances of planets from the Sun can vary slightly depending on the source, as they may use different methods of measurement or provide updated data.

If you have specific distances from the article by William A. Gutsch Jr. that you would like to use, please provide them here. I can then help you make calculations based on the distances you share.

Here is some info that will help:

1. Mercury is the closest planet to the sun. Its average distance from the sun is about 35.9 million miles (57.8 million kilometers)

2. Your next stop is Venus. This planet travels around the sun in a nearly circular orbit, at an average distance of about 67.2 million miles (108.2 million kilometers). Venus takes 224.7 Earth days to make one trip around the sun. It is nearly the same size as the Earth, and its diameter is about 7,519 miles

3. Beyond the orbit of Venus is Earth. The Earth's orbit is almost as circular as that of Venus. The Earth travels around the sun at an average distance of 93 million miles (150 million kilometers). It completes one orbit around the sun in 365.25 days. It takes 23 hours and 56 minutes for the Earth to rotate on its axis.

4. Your next stop is Mars, a planet that shines in the sky with a reddish color and is sometimes called the Red Planet. The orbit of Mars is more oval-shaped than that of the Earth. So the distance between Mars and the sun may be as little as 128.4 million miles (206.7 million kilometers) or as much as 154.8 million miles (249.2 million kilometers). The average distance between Mars and the sun is about 142 million miles (228 million kilometers).

5. After passing through a belt of asteroids beyond the orbit of Mars, the next stop on your planetary journey is Jupiter. This planet is more than five times farther from the sun than Earth. Jupiter travels around the sun at an average distance of 483 million miles (778 million kilometers). It takes 11.9 Earth years to complete one orbit.

6. The next stop on your journey is Saturn, another gas giant. It orbits the sun at an average distance of 885.5 million miles (1.4 billion kilometers). It takes nearly 29.5 Earth years to make one trip around the sun.

7. Leaving Saturn and traveling even farther from the sun, you come to the planet Uranus. Uranus is 32,000 miles (51,500 kilometers) in diameter. It orbits the sun at an average distance of 1.8 billion miles (2.9 billion kilometers). It takes about 84 Earth years to complete one orbit. Uranus rotates once on its axis in an east-to-west direction in approximately 17 hours.

8. Your last stop is Neptune Neptune's average distance from the sun is about 2.8 billion miles (4.5 billion kilometers). It takes nearly 165 Earth years to complete one orbit. Its diameter is 30,800 miles (49,600 kilometers).

Thank you for providing the distances from the article. Using the data you've shared, here are the average distances of each planet from the Sun converted to scientific notation, along with the calculations based on that information:

a. List of Distances from Each Planet to the Sun (in Scientific Notation):

1. Mercury: 35.9 million miles = \(3.59 \times 10^{7}\) miles
2. Venus: 67.2 million miles = \(6.72 \times 10^{7}\) miles
3. Earth: 93 million miles = \(9.3 \times 10^{7}\) miles
4. Mars: 142 million miles = \(1.42 \times 10^{8}\) miles
5. Jupiter: 483 million miles = \(4.83 \times 10^{8}\) miles
6. Saturn: 885.5 million miles = \(8.855 \times 10^{8}\) miles
7. Uranus: 1.8 billion miles = \(1.8 \times 10^{9}\) miles
8. Neptune: 2.8 billion miles = \(2.8 \times 10^{9}\) miles

b. Addition of Two Distances

Let's add the distances of Earth and Mars.

\[ 9.3 \times 10^{7} , \text{miles} + 1.42 \times 10^{8} , \text{miles} = (9.3 + 14.2) \times 10^{7} = 23.5 \times 10^{7} = 2.35 \times 10^{8} , \text{miles} \]

c. Subtraction of Two Distances

Let's subtract the distance of Mercury from the distance of Saturn.

\[ 8.855 \times 10^{8} , \text{miles} - 3.59 \times 10^{7} , \text{miles} = 8.855 \times 10^{8} - 0.359 \times 10^{8} = (8.855 - 0.359) \times 10^{8} = 8.496 \times 10^{8} , \text{miles} \]

d. Multiplication of Two Distances

Let's multiply the distances of Venus and Jupiter.

\[ (6.72 \times 10^{7}, \text{miles}) \times (4.83 \times 10^{8}, \text{miles}) = (6.72 \times 4.83) \times 10^{15} \approx 32.45 \times 10^{15} = 3.245 \times 10^{16} , \text{miles}^2 \]

e. Division of Two Distances

Let's divide the distance of Neptune by the distance of Uranus.

\[ \frac{2.8 \times 10^{9} , \text{miles}}{1.8 \times 10^{9} , \text{miles}} = \frac{2.8}{1.8} \approx 1.555 \implies 1.555 \]

Summary of Calculations

1. a. Average distances of planets are now in scientific notation.
2. b. Earth + Mars = \(2.35 \times 10^{8}\) miles.
3. c. Saturn - Mercury = \(8.496 \times 10^{8}\) miles.
4. d. Venus × Jupiter = \(3.245 \times 10^{16}\) miles².
5. e. Neptune ÷ Uranus ≈ \(1.555\).

If you have any further questions or need additional calculations, feel free to ask!

Can you do another one with D. ?

Certainly! Let's perform another multiplication using two different distances.

This time, let’s multiply the distances of Earth and Saturn.

d. Multiplication of Two Distances

The distances in scientific notation are:

• Earth: \(9.3 \times 10^{7}\) miles
• Saturn: \(8.855 \times 10^{8}\) miles

Now, we will multiply these distances:

\[ (9.3 \times 10^{7} , \text{miles}) \times (8.855 \times 10^{8} , \text{miles}) \]

Step 1: Multiply the coefficients:

\[ 9.3 \times 8.855 \approx 82.305 \]

Step 2: Add the exponents (since both terms have powers of 10):

\[ 10^{7} \times 10^{8} = 10^{(7 + 8)} = 10^{15} \]

Step 3: Combine the results:

\[ 82.305 \times 10^{15} \]

To express this in proper scientific notation, we convert \(82.305\) to \(8.2305 \times 10^{1}\) and adjust the exponent:

\[ 82.305 \times 10^{15} = 8.2305 \times 10^{16} \]

Final Result

So, the product of the distances of Earth and Saturn is:

\[ 8.2305 \times 10^{16} , \text{miles}^2 \]

If you need any further calculations or assistance, feel free to ask!

Can you elaborate more on E. ?

Absolutely! Let’s delve deeper into part e, where we will divide the distance of Neptune by the distance of Uranus.

e. Division of Two Distances

Distances in Scientific Notation:

• Neptune: \(2.8 \times 10^{9}\) miles
• Uranus: \(1.8 \times 10^{9}\) miles

Step-by-step Calculation:

1. Set up the division:

   \[ \frac{2.8 \times 10^{9} , \text{miles}}{1.8 \times 10^{9} , \text{miles}} \]

2. Divide the coefficients:

   We need to divide the numeric values (the coefficients):

   \[ \frac{2.8}{1.8} \approx 1.5556 \]

3. Divide the powers of ten:

   When you divide terms with the same base (in this case, \(10^{9}\)), you can cancel out the exponents:

   \[ \frac{10^{9}}{10^{9}} = 10^{(9-9)} = 10^{0} = 1 \]

4. Combine the results:

   Therefore, the result of the division is simply the coefficient divided by the powers of ten:

   \[ \frac{2.8 \times 10^{9}}{1.8 \times 10^{9}} = 1.5556 \times 10^{0} = 1.5556 \]

Final Result:

The result of dividing the distance of Neptune by the distance of Uranus is:

\[ \approx 1.56 \]

This indicates that Neptune is approximately 1.56 times farther from the Sun compared to Uranus based on the distances provided.

If you have any more questions or if you'd like me to do another operation or provide more details on a specific topic, feel free to let me know!

4. compare and contrast adding/subtracting numbers written in scientific notation with multiplying/dividing numbers written in scientific notation. How are the steps in each cace similar? how are they different? do you think adding, subtracting, multiplying, or dividing numbers written in scientific notation is easiest? why?

Adding and subtracting numbers written in scientific notation, as well as multiplying and dividing them, involve different processes and steps. Below is a comparison of the two approaches.

Adding and Subtracting in Scientific Notation

Steps for Addition and Subtraction:

1. Match Exponents: Before adding or subtracting, the numbers must have the same exponent. If the exponents are different, you'll need to adjust one of the numbers to match the other by changing its coefficient and exponent accordingly.

   • For example, to add \(3.0 \times 10^5\) and \(4.5 \times 10^6\), you could first express \(3.0 \times 10^5\) as \(0.30 \times 10^6\) to match the exponents.

2. Add or Subtract Coefficients: Once the exponents are the same, you simply add or subtract the coefficients.

   • Continuing with the previous example, \(0.30 \times 10^6 + 4.5 \times 10^6 = (0.30 + 4.5) \times 10^6 = 4.8 \times 10^6\).

3. Final Result: Express the result in proper scientific notation if necessary.

Multiplying and Dividing in Scientific Notation

Steps for Multiplication and Division:

1. Multiply or Divide Coefficients: For multiplication, you multiply the coefficients, and for division, you divide the coefficients.

   • Example: For \(2.0 \times 10^3 \cdot 4.0 \times 10^2\), you'd calculate \(2.0 \cdot 4.0 = 8.0\).

2. Add or Subtract Exponents: For multiplication, you add the exponents. For division, you subtract the exponents.

   • Example: Continuing with the multiplication example, \(10^3 \cdot 10^2 = 10^{(3+2)} = 10^5\).

3. Combine: The result combines the new coefficient with the new power of ten. In our example, \(8.0 \times 10^5\).

4. Final Result: Again, express the final result in proper scientific notation.

Similarities

• Both processes require precision in working with coefficients and exponents.
• Regardless of whether you're adding/subtracting or multiplying/dividing, it's important to express the final answer in proper scientific notation.
• Both methods can involve adjustments to the coefficients and exponents to simplify calculations or ensure proper notation.

Differences

• Exponents: In addition and subtraction, matching exponents is crucial. In multiplication and division, you manipulate the exponents by either adding (in multiplication) or subtracting (in division) them directly.
• Operations: Addition and subtraction focus purely on the coefficients after adjusting the exponents, while multiplication and division involve both coefficients and exponents in their operations.

Easiest Operation

In terms of ease, opinions may vary:

• Multiplication and Division: Many find these operations easier because you can work directly with the coefficients and the exponents independently, allowing for clear calculations and reduction.

• Addition and Subtraction: These may be perceived as slightly more cumbersome due to the need to match the exponents first, particularly when the numbers differ significantly in magnitude.

Conclusion: Overall, the perspective on which operation is easiest can depend on individual comfort with mathematics and familiarity with scientific notation. Personally, I find multiplication and division easier due to the independent handling of coefficients and exponents, while the requirement to adjust exponents in addition and subtraction can be an extra step that complicates the process.