Final answer:
The average distance of the asteroid from the sun is 600 million kilometers. The exact time it takes for the asteroid to orbit the sun cannot be determined without knowing the constant k in Kepler's Third Law. However, it will be shorter than 8 years.
Step-by-step explanation:
The average distance an asteroid orbits the sun is called its semi-major axis. In this case, the asteroid has an average distance of 4 astronomical units (AU) from the sun. 1 AU is defined as the average distance between the Earth and the Sun, which is about 150 million kilometers. So, the average distance of the asteroid from the sun would be 4 times 150 million kilometers, which is 600 million kilometers.
Now, to calculate the time it takes for the asteroid to orbit the sun, we can use Kepler's Third Law of Planetary Motion. This law states that the square of the period of an object's orbit is proportional to the cube of its semi-major axis.
Let's assume the period of the asteroid's orbit is represented by T. Using the formula T^2 = k(a^3), where T is the period, a is the semi-major axis, and k is a constant, we can solve for the period:
- Substitute the given values: T^2 = k(4^3)
- Simplify: T^2 = 64k
- Take the square root of both sides: T = √(64k)
The exact value of the constant k will depend on the units used, but we can disregard it for now. Therefore, the period T is equal to the square root of 64 times the constant k.
So, to calculate the exact period of the asteroid's orbit, we would need to know the value of the constant k. Without that information, we can't determine the exact time it takes for the asteroid to orbit the sun. However, we can say that the period will be proportional to the square root of 64, meaning it will be shorter than the square root of 64 years, which is 8 years.