To determine Earth’s distance from the Sun in July, we will first analyze the information given about Earth's elliptical orbit.
- 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.
- 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.
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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).
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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: \[ c = \sqrt{22350.25 - 22344} \approx \sqrt{6.25} \approx 2.5 \text{ million kilometers} \]
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The Sun, located at the focus, is approximately 2.5 million kilometers from the center of the ellipse.
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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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