Final answer:
The diameter of Earth at 1.3 x 10^7 meters is 109 times smaller than the Sun's diameter and 7.45 x 10^8 times smaller than a light-year. In a scale model of the solar system with a reduction factor of 10^9, Earth would be about 1.3 cm in diameter. Earth's surface area converts to 1.69 x 10^8 km² when calculated from its diameter in kilometers.
Step-by-step explanation:
To compare the diameter of Earth with other measurements, we start with the fact that Earth's diameter is approximately 1.3 x 10^7 meters (12,700 km). For instance, the Sun's diameter is around 1.39 million kilometers, which is about 109 times the diameter of Earth. When considering the vastness of space, a light-year, which is the distance light travels in one year, equates to about 9.46 × 10¹² kilometers. Converting this to Earth diameters, we find that 1 light-year is approximately 7.45 × 10^8 Earth diameters. This is equal to about 745 million times the diameter of Earth.
When imagining a scale model of the solar system with a scale factor of 1 billion (10^9), Earth's diameter would shrink to about the size of a grape, or 1.3 centimeters. The Earth-Moon system could fit into a standard backpack, with the Moon represented by a pea orbiting the grape-sized Earth at a distance of 40 centimeters.
To convert Earth's surface area to square kilometers, we would either convert the diameter from meters to kilometers and square it, or convert the surface area directly. Starting with Earth's diameter of 1.3 x 10^7 meters, we would find that in kilometers, this is 13,000 km. Squaring that to find the surface area gives us 1.69 x 10^8 km², which matches the order of magnitude of 10¹´ m² when converted to square kilometers.