Earth orbits the sun in an elliptical pattern. the equation of earths path is (x+2.5)^2/22350.25 + y^2/22344=1 where the measurements represent millions of kilometers. The sun is located at the focus (0,0). In january, earth id located at one vertices which is 147 million kilometers from the sun. Determine earths distance from the sun in July.
In July, the earth will be at the other vertex.
Since the equation can be written as
(x+2.5)^2/149.5^2 + y^2/149.479^2
you know that the major axis has length 299 million km
so, how far is the other vertex from (0,0)?
To determine Earth's distance from the sun in July, we first need to understand the equation that represents Earth's path. The equation of Earth's path is given as:
(x + 2.5)^2 / 22350.25 + y^2 / 22344 = 1
Here, x and y represent the coordinates on the Cartesian plane, with the sun located at the origin (0,0). The values 22350.25 and 22344 are coefficients used to determine the shape and size of the elliptical orbit.
In January, Earth is located at one of the vertices of the ellipse, which is 147 million kilometers away from the sun. We can use this known value to find the other vertex, which represents Earth's position in July.
Let's solve for the distance from the sun in July:
(x + 2.5)^2 / 22350.25 + y^2 / 22344 = 1
In January, x = 147, so we can substitute this into the equation:
(147 + 2.5)^2 / 22350.25 + y^2 / 22344 = 1
(149.5)^2 / 22350.25 + y^2 / 22344 = 1
Now, we need to solve for y. Rearranging the equation:
y^2 / 22344 = 1 - (149.5)^2 / 22350.25
y^2 / 22344 = 1 - 149.5^2 / 22350.25
y^2 / 22344 = (22350.25 - 149.5^2) / 22350.25
y^2 / 22344 = 22350.25 - 149.5^2 / 22350.25
y^2 / 22344 = 22296.0025 / 22350.25
Now, we can solve for y:
y^2 = (22296.0025 / 22350.25) * 22344
y^2 = 22296.0025
Taking the square root of both sides:
y = sqrt(22296.0025)
y ≈ 149.323
So, the distance from the sun to Earth in July is approximately 149.323 million kilometers.