if earth is one astronomical unit from the sun and has a period of 1 year, approximately how far is planet X form the sun if it has a period of 26 yearS
Kepler's third law says that R^3/P^2 = constant.
http://csep10.phys.utk.edu/astr161/lect/history/kepler.html
We can choose to measure P in years and the orbital radius R in a.u.
In your case, the ratio (R^3/26^2) = 1^3/1^2 = 1
R^3 = 676
R = 8.8 a.u.
To determine the approximate distance of planet X from the sun, we can use Kepler's Third Law of planetary motion, which states that the square of the orbital period of a planet is proportional to the cube of its average distance from the sun.
Let's denote the average distance of planet X from the sun as "d" (in astronomical units, AU). We can set up the following equation:
(Period of Earth / Period of Planet X)^2 = (Average Distance of Earth / Average Distance of Planet X)^3
We can plug in the values:
(1 year / 26 years)^2 = (1 AU / d)^3
Simplifying, we have:
(1/26)^2 = 1/d^3
1/676 = 1/d^3
To isolate "d^3", we can take the reciprocal of both sides:
d^3 = 676
Taking the cube root of both sides, we find:
d ≈ ∛676
Calculating the cube root of 676, we get:
d ≈ 8.42
Therefore, the average distance of planet X from the sun is approximately 8.42 astronomical units (AU).
To determine the approximate distance of planet X from the sun, you can make use of Kepler's Third Law of Planetary Motion, which states that the square of the orbital period of a planet is directly proportional to the cube of its average distance from the sun.
Let's denote the distance of planet X from the sun as "d". Since the orbital period of planet X is 26 years and the Earth's orbital period is 1 year, we can set up the following proportion:
(26 years)^2 / d^3 = (1 year)^2 / (1 AU)^3
Before solving for "d," let's simplify the equation by plugging in the values:
(26^2) / d^3 = (1^2) / (1 AU)^3
Simplifying further:
26^2 = d^3
676 = d^3
To find the cube root of 676, we get:
d ≈ ∛676
Using a calculator, you'll find that the approximate value of ∛676 is 8.38. Therefore, the approximate distance of planet X from the sun would be around 8.38 astronomical units (AU).