Answer :
Answer:
121.54 million mi.
Step-by-step explanation:
Ellipses
An ellipse is an oval-like shape where all the points along the ellipse's shape have the same summed distance from it to each focus.
The equation is this
[tex]\dfrac{(x-h)^2}{a^2} +\dfrac{(y-k)^2}{b^2} =1[/tex],
where
- a is the distance between the center and one of the vertices along the major axis
- b is the distance between the center and one of the vertices along the minor axis
- h and k are the x and y values of the ellipse's center
Every ellipse has a major and minor axis parallel to the x and y axis; the major axis is the largest diameter of the shape and the minor axis is the smallest.
The distance between the center and one of the foci is c. The values of c and a can be used to find the value of b using the Pythagorean Theorem [tex]b=\sqrt{a^2-c^2}[/tex] (a > c).
[tex]\dotfill[/tex]
Eccentricity
The eccentricity of an ellipse measures how "elliptical" the given ellipse is. (An eccentricity between 0 and 1 is indicates an ellipse).
The formula is
[tex]e=\dfrac{c}{a}[/tex],
where c is the distance from the ellipse's center to either foci and a is half the distance of the major axis (the longest length of the ellipse).
[tex]\hrulefill[/tex]
Solving the Problem
Visualize the Problem
We can start by drawing an ellipse that represents the comet's trajectory.
Drawing two foci inside the ellipse, we label one of them as the sun.
If the comet that's along the ellipse has the greatest distance of 2515 million mi from the sun then it must be at the vertice--along the major axis-- opposite to where the sun is located.
The distance between them on the image is half the length of the major axis which is a, plus the distance from the sun to the center of the ellipse, c--connecting to that half length mentioned earlier--.
This also means that the shortest distance between the comet and the sun is the distance between the vertice near the sun, which is the difference between a and c.
(See the image attached).
[tex]\dotfill[/tex]
Solving for a - c
Using all the information given in the problem and from the image we've made we know,
- [tex]e=0.9078=\dfrac{c}{a}[/tex]
- [tex]a+c=2515[/tex]
and we need to find a - c.
This is just a case of solving a set of equations!
We can write c in terms of a from the first equation,
[tex]0.9078a=c[/tex].
Plugging it into the second equation we can get the value of a!
[tex]a+0.9078a=2515[/tex]
[tex]1.9078a=2515[/tex]
[tex]a=\dfrac{2515}{1.9078}=1318.27[/tex]
Plugging it back in into the second equation, the value of c is
[tex]1318.27+c=2515[/tex]
[tex]c=2515-1318.27=1196.73[/tex].
Our values are right since the sum of 1196.73 and 1318.27 does equal 2515.
So, a - c = 1317.27 - 1196.73 = 121.54 million mi.
