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
SUB TO YFBG JAY ASAP
AleksAgata [21]

Answer:

I done did it.

Step-by-step explanation:

3 0
2 years ago
Read 2 more answers
HELP ME PLEASE I WILL GIVE BRAINLEST HELP
ahrayia [7]

Answer:

Which one you need me to answer

Step-by-step explanation:

8 0
2 years ago
Find a polynomial of degree 3 with real cofficients and zeros of -3,-1,4 for which f(-2)=24​
konstantin123 [22]

\bf \textit{zeros at } \begin{cases} x = -3\implies &x+3=0\\ x = -1\implies &x+1=0\\ x = 4\implies &x-4=0 \end{cases}\qquad \implies (x+3)(x+1)(x-4)=\stackrel{y}{0} \\\\\\ (x^2+4x+3)(x-4)=0\implies x^3~~\begin{matrix}+ 4x^2 \\[-0.7em]\cline{1-1}\\[-5pt]\end{matrix}~~+3x~~\begin{matrix} -4x^2 \\[-0.7em]\cline{1-1}\\[-5pt]\end{matrix}~~-16x-12=0 \\\\\\ x^3-13x-12=0

we know that f(-2) = 24, namely when x = -2, y = 24, let's see if that's true

\bf x^3-13x-12=y\implies \stackrel{x = -2}{(-2)^3-13(-2)-12}=y \\\\\\ -8+26-22=y \implies 6=y

darn!! no dice.... hmmmm wait a second.... 4 * 6 = 24, if we could just use a common factor of 4 on the function, that common factor times 6 will give us 24, let's check.

\bf 4(x^3-13x-12)=y\implies \stackrel{x = -2}{4[~~(-2)^3-13(-2)-12~~]}=y \\\\\\ 4[~~-8+26-22~~]=y\implies 4[6]=y\implies 24=y \\\\[-0.35em] ~\dotfill\\\\ ~\hfill 4x^3-52x-48=y~\hfill

4 0
3 years ago
isaac has a piece of rope that is 5 yards long. into how many 1/2 yard pieces of rope can isaac cut the rope
marusya05 [52]
What you have to do is figure out how many half inches you need to have to figure out how many half inches equal 5 yards. Therefore if there are 12 inches in a foot, then that would be 24 half inches. So..... three feet equal 1 yards. So multiply 3 by 24. 3x24=72 so you would need 72 half inches 
8 0
3 years ago
If it takes Tim 20 minutes to bike 6 miles, how many minutes will it take him to bike 144 miles?​
kari74 [83]

Answer:

144/6 = 24

24 x 20 = 480

480 is the answer

3 0
2 years ago
Read 2 more answers
Other questions:
  • What is 5x-15x + 20y = x + 5<br> Solve for y
    5·2 answers
  • Find the sum of the even integers between 21 and 45
    7·2 answers
  • X f(x)
    5·1 answer
  • A rectangle has a perimeter of 48 feet and a length of 14 ft which equation can you solve to find the width
    6·1 answer
  • 1. Rename 4/7 as a percent. Round to the nearest tenth of a percent if necessary
    15·2 answers
  • Please Answer questions 29 thru 40 for me please it's due tommorow!!
    12·1 answer
  • I'll give brainlist if anyone answer correctly on these 2 questions
    14·2 answers
  • The table shows Mitch’s record for the last thirty par-3 holes he has played.
    7·1 answer
  • 2x + 2y = -2<br> 3x - 2y = 12<br> Answer:
    9·1 answer
  • What is cos 0 when sin 0 = 2/5
    14·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!