1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
Sholpan [36]
3 years ago
13

The planets in our solar system do not travel in circular paths. Rather, their orbits are elliptical. The Sun is located at a fo

cus of the ellipse.
The perihelion is the point in a planet’s orbit that is closest to the Sun. So, it is the endpoint of the major axis that is closest to the Sun.
The aphelion is the point in the planet’s orbit that is furthest from the Sun. So, it is the endpoint of the major axis that is furthest from the Sun.
The closest Mercury comes to the Sun is about 46 million miles. The farthest Mercury travels from the Sun is about 70 million miles.
1. What is the distance between the perihelion and the aphelion?
2. What is the distance from the center of Mercury’s elliptical orbit and the Sun?
3. Write the equation of the elliptical orbit of Mercury, where the major axis runs horizontally. Allow a and b to be measured in millions of miles. Use the origin as the center of the ellipse.
4. What is the eccentricity of the ellipse? Round your answer to the nearest thousandth.
5. What does the value of the eccentricity tell you about the relative shape of the ellipse?
Mathematics
1 answer:
qwelly [4]3 years ago
5 0

1. The distance between the perihelion and the aphelion is 116 million miles

2. The distance from the center of Mercury’s elliptical orbit and the Sun is 12 million miles

3. The equation of the elliptical orbit of Mercury is \frac{x^{2}}{3364}}+\frac{y^{2}}{3220}=1

4. The eccentricity of the ellipse is 0.207 to the nearest thousandth

5. The value of the eccentricity tell you that the shape of the ellipse is near to the shape of the circle

Step-by-step explanation:

Let us revise the equation of the ellipse is

\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 , where the major axis is parallel to the x-axis

  • The length of the major axis is 2a
  • The coordinates of the vertices are (± a , 0)
  • The coordinates of the foci are (± c , 0) , where c² = a² - b²

∵ The Sun is located at a focus of the ellipse

∴ The sun located ate c

∵ The perihelion is the point in a planet’s orbit that is closest to the

   Sun ( it is the endpoint of the major axis that is closest to the Sun )

∴ The perihelion is located at the vertex (a , 0)

∵ The closest Mercury comes to the Sun is about 46 million miles

∴ The distance between a and c is 46 million miles

∵ The aphelion is the point in the planet’s orbit that is furthest from

   the Sun ( it is the endpoint of the major axis that is furthest from

   the Sun )

∴ The aphelion is located at the vertex (-a , 0)

∵ The farthest Mercury travels from the Sun is about 70 million miles

∴ The distance from -a to c is 70 million miles

∴ The distance between the perihelion and the aphelion =

   70 + 46 = 116 million miles

1. The distance between the perihelion and the aphelion is 116 million miles

∵ The distance between the perihelion and the aphelion is the

  length of the major axis of the ellipse

∵ The length of the major axis is 2 a

∴ 2 a = 116

- Divide both sides by 2

∴ a = 58

∴ The distance from the center of Mercury’s elliptical orbit to the

   closest end point to the sun is 58 million miles

∵ The distance between the sun and the closest endpoint is

   46 million miles

∴ The distance from the center of Mercury’s elliptical orbit and

   the Sun = 58 - 46 = 12 million miles

2. The distance from the center of Mercury’s elliptical orbit and the Sun is 12 million miles

∵ The major axis runs horizontally

∴ The equation is \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1

∵ a = 58

∵ c is the distance from the center to the focus of the ellipse

∴ c = 12

∵ c² = a² - b²

∴ (12)² = (58)² - b²

- Add b² to both sides

∴ (12)² + b² = (58)²

- Subtract (12)² from both sides

∴ b² = (58)² - (12)² = 3220

- Substitute these values in the equation

∴ \frac{x^{2}}{3364}}+\frac{y^{2}}{3220}=1

3. The equation of the elliptical orbit of Mercury is \frac{x^{2}}{3364}}+\frac{y^{2}}{3220}=1

The eccentricity (e) of an ellipse is the ratio of the distance from the

center to the foci (c) and the distance from the center to the

vertices (a) ⇒ e=\frac{c}{a}

∵ c = 12

∵ a = 58

∴ e=\frac{12}{58} = 0.207

4. The eccentricity of the ellipse is 0.207 to the nearest thousandth

If the eccentricity is zero, it is not squashed at all and so remains a circle.

If it is 1, it is completely squashed and looks like a line

∵ The eccentricity of the ellipse is 0.207

∵ This number is closed to zero than 1

∴ The shape of the ellipse is near to the shape of the circle

5. The value of the eccentricity tell you that the shape of the ellipse is near to the shape of the circle

Learn more:

You can learn more about conics section in brainly.com/question/4054269

#LearnwithBrainly

You might be interested in
Not sure how to do this
Sergio039 [100]
45 i believe it is i may be wrong!!!
5 0
2 years ago
What is the answer to 5 + 7x – (2x + 3)
VikaD [51]

Answer:

2+5x?

Step-by-step explanation:

3 0
3 years ago
The answer of this question
dolphi86 [110]

Answer:

i think it is c

Step-by-step explanation:

not 100% sure tho

4 0
3 years ago
Which angle is in standard position
Readme [11.4K]

Answer:

The bottom one

Step-by-step explanation:

its vertex is located at the origin and one ray is on the positive x-axis

4 0
3 years ago
Read 2 more answers
6x + 5x + 2y <br> 11x + 2y <br> 13xy <br> Explain what is incorrect in the problem above?
weqwewe [10]

Answer:

Step-by-step explanation:

6x +5x +2y and what i did is that i simplify and i got

11x + 2y

The expression is not factorable with rational numbers.

the thing is you can't go any farther than that

8 0
3 years ago
Other questions:
  • Caleb scored 60 on his first math test. He studied diligently for the next test and earned a score of 90. What is his percent of
    7·1 answer
  • A roller coaster at six flags is 1.5 kilometers long. how many miles is it?
    5·1 answer
  • A square pyramid has a height of 12 units and a volume of 256 units3. If a square prism has the same the base area and volume as
    10·1 answer
  • What is the value of x, given that pq is parallel to bc?
    15·1 answer
  • Pls help me get full marks
    8·2 answers
  • How much more would $5,218.10 earn in 6 years, compounded daily at 3.8%, when compared to the interest on $5,218.10 over 6 years
    14·1 answer
  • Help me pls. Which decimal is less than 0.8?
    12·1 answer
  • The graph of y=f(x) is shown below. Which point could be used to find f(-2)
    12·1 answer
  • A tennis ball has a radius of about 1.4 inches What is the volume of the tennis ball to the nearest half inch.? Use 3.14 as an a
    7·1 answer
  • Solve for m.<br> m/-3 = 7
    5·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!