Lost is an understatement..please help me understand this.
Orbits and Distance Johannes Kepler (1571–1630)discovered a relationship between a planet’s distance
D from the sun and the time T it takes to orbit the sun. This formula is , where T is in
Earth years and corresponds to the distance
between Earth and the sun, or 93,000,000 miles.
(a) Neptune is 30 times farther from the sun than
Earth . Estimate the number of years
required for Neptune to orbit the sun.
(b) Write this formula with rational exponents
T = k D^(3/2)
let x = 93,000,000 miles
for earth
1 year = k x^(3/2)
k = 1/x^(3/2)
for Neptune
T = [1/x^3/2) (30x)^(3/2)
T = 30^(3/2) = 164 years
To understand the relationship between a planet's distance and the time it takes to orbit the sun, let's break it down step by step.
(a) To estimate the number of years required for Neptune to orbit the sun, we need to use the formula discovered by Johannes Kepler:
T = D^(3/2)
For this formula, T represents the time it takes for a planet to orbit the sun (in Earth years), and D represents the distance between the planet and the sun (in astronomical units).
We are given that Neptune is 30 times farther from the sun than Earth. If we assume the distance between Earth and the sun is 1 astronomical unit, then the distance between Neptune and the sun would be 30 astronomical units (since it is 30 times farther).
Now, we can substitute this value into the formula:
T = (30)^(3/2)
T = 30^(3/2)
T ≈ 195.35 Earth years
Therefore, it would take Neptune approximately 195.35 Earth years to orbit the sun.
(b) To write the formula with rational exponents, we can rewrite it as:
T = D^(3/2)
This formula shows that the time it takes for a planet to orbit the sun is equal to the distance between the planet and the sun raised to the power of 3/2.