A diagram of Earth's orbit as an ellipse is shown.A horizontal ellipse is shown with left vertex labeled with "Earth's position in July" and right vertex labeled "Earth's position in January. The sun is located at the focus right of center. The distance from the sun to the right vertex is labeled "147,000,000 km". The distance from the sun to the left vertex is labeled "? km". The diagram is not drawn to scale.

Question
Use the image to answer the question.
Earth orbits the sun in an elliptical pattern. The equation of Earth’s path is start fraction left parenthesis x plus 2.5 right parenthesis squared over 22350.25 end fraction plus start fraction y squared over 22344 end fraction equals 1 where the measurements represent millions of kilometers. The sun is located at the focus whose coordinate is left parenthesis 0 comma 0 right parenthesis. In January, Earth is located at one of the vertices, which is 147 million kilometers from the sun. Determine Earth’s distance from the sun in July.
(1 point)
Responses

147 million kilometers
147 million kilometers

149.5 million kilometers
149.5 million kilometers

152 million kilometers
152 million kilometers

154.5 million kilometers

1 answer

To determine Earth’s distance from the Sun in July, we will first analyze the information given about Earth's elliptical orbit.

  1. The distance from the Sun to the Earth's position in January is 147 million kilometers. This is located at one of the vertices of the ellipse.
  2. Since the equation of the ellipse is given in the standard form, we can identify aspects crucial for calculating distances.

From the equation of the ellipse: \[ \frac{(x + 2.5)^2}{22350.25} + \frac{y^2}{22344} = 1 \]

  • This can be compared to the standard form of an ellipse which is: \[ \frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1 \] where \((h, k)\) is the center, \(a\) is the semi-major axis, and \(b\) is the semi-minor axis.
  1. From the equation, we have \(a^2 = 22350.25\) and \(b^2 = 22344\).

    • Therefore, \(a = \sqrt{22350.25} \approx 149.5\) million kilometers (since the units are in millions of kilometers).
    • \(b = \sqrt{22344} \approx 149.48\) million kilometers (similarly).
  2. The focus of the ellipse is at \(c\) distance from the center, where \(c\) is given by: \[ c = \sqrt{a^2 - b^2} \] Calculating it: \[ c = \sqrt{22350.25 - 22344} \approx \sqrt{6.25} \approx 2.5 \text{ million kilometers} \]

  3. The Sun, located at the focus, is approximately 2.5 million kilometers from the center of the ellipse.

  4. However, the vertex at January indicates that we are at the farthest point from the Sun— the perihelion. The other vertex (July) will be at: \[ 147 \text{ million km} + 2 \times c = 147 + 2 \cdot 2.5 \text{ million km} \] Since the distance between the two vertices is \(2a = 2 \cdot 149.5\):

    • The required distance (the aphelion) from the Sun in July is approximately \(147 + 2 \cdot 2.5 \approx 152\) million kilometers.

Therefore, Earth's distance from the sun in July is approximately 152 million kilometers.

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