The Earth's orbit is an ellipse with the sun at one focus. The length of the major axi is 186,000,000 miles and the accentricity is 0.0167. Find the distance from the ends of the major axis to the sun. These are the greatest and least distances from the earth to the sun.
You have a = 93 (million miles)
e = c/a = 0.0167
so, c = 1.55
now just find a±c
a = 93,000,000 miles.
eccentricity = 0.0167
c/a = 167/10000
c = 167/10000 * 93,000,000 = 1,553,100
c^2 = a^2 - b^2
solve for b
make a sketch to see what calculations you need.
To find the greatest and least distances from the Earth to the Sun, we need to calculate half the length of the major axis, also known as the semi-major axis (a), and use it together with the eccentricity (e).
Given:
Major axis length (2a) = 186,000,000 miles
Eccentricity (e) = 0.0167
Step 1: Find the semi-major axis (a)
The semi-major axis (a) is half the major axis length, so we divide it by 2:
a = (186,000,000 miles) / 2
a = 93,000,000 miles
Step 2: Find the distance from the sun to the ends of the major axis
We can calculate the distances using the formula:
Greatest distance (d1) = a * (1 + e)
Least distance (d2) = a * (1 - e)
Calculating the greatest distance:
d1 = 93,000,000 miles * (1 + 0.0167)
d1 ≈ 93,000,000 miles * 1.0167
d1 ≈ 94,622,100 miles
Calculating the least distance:
d2 = 93,000,000 miles * (1 - 0.0167)
d2 ≈ 93,000,000 miles * 0.9833
d2 ≈ 91,065,900 miles
Therefore, the greatest distance from the Earth to the Sun is approximately 94,622,100 miles, and the least distance is approximately 91,065,900 miles.
To find the greatest and least distances from the Earth to the Sun, we first need to understand the different components of an ellipse.
An ellipse is a geometric shape that can be defined by its major and minor axes. The major axis is the longest diameter of the ellipse, and the minor axis is the shortest diameter.
In this case, we are given the length of the major axis, which is 186,000,000 miles. The major axis typically represents the longest distance between any two points on the ellipse.
The eccentricity of an ellipse determines how elongated or circular the shape is. It is a measure of how much the ellipse deviates from a perfect circle. The eccentricity value ranges from 0 to 1, with 0 representing a perfect circle and 1 representing a highly elongated ellipse.
In this case, the eccentricity is given as 0.0167, indicating that the Earth's orbit is slightly elongated.
To find the greatest and least distances from the Earth to the Sun, we can use the formula:
Distance from the ends of the major axis to the Sun = (1 ± eccentricity) × (Length of major axis / 2)
Let's calculate the greatest and least distances:
Greatest distance from the Earth to the Sun = (1 + 0.0167) × (186,000,000 / 2)
= 1.0167 × 93,000,000
= 94,494,100 miles
Least distance from the Earth to the Sun = (1 - 0.0167) × (186,000,000 / 2)
= 0.9833 × 93,000,000
= 91,128,900 miles
Therefore, the greatest distance from the Earth to the Sun is approximately 94,494,100 miles, and the least distance is approximately 91,128,900 miles.