How many years would it take the planet to orbit the sun. How do you calculate. All I have is, that the planet is 97 times farther from sun than earth or 97 Astronomical units

Using Kepler's Law you find that the anser is 97^(1.5) years. You can estimate this without using a calculator:

97^(1.5) = (100-3)^(1.5) =
[100*(1 - 3/100)]^(1.5) =
100^(1.5)*(1-3/100)^(1.5)=

1000*[1-1.5*3/100 + (1.5)*(0.5)/2 (3/100)^2 + term of order (3/100)^3] =

1000 - 45 + 27/80 + term of order 1/100

27/80 is about 1/3, so the answer is
955.3 years.

To calculate the number of years it would take for a planet to orbit the sun, you can use Kepler's Third Law of Planetary Motion. This law states that the square of the orbital period of a planet is directly proportional to the cube of its average distance from the sun.

In this case, you mentioned that the planet is 97 times farther from the sun than Earth, or 97 astronomical units (AU). So, we can set up the equation as follows:

(T/1 year)^2 = (97 AU)^3

To solve this, you'll need to take the square root of both sides:

T/1 year = √(97 AU)^3

Now, you can simplify the expression by taking the square root of both sides:

T/1 year = (97 AU)^(3/2)

Since you mentioned that you can estimate this without using a calculator, we can use approximation techniques.

Expressing 97 as (100-3), we can rewrite the equation as:

T/1 year = (100-3)^(3/2)

Expanding this using binomial expansion, we get:

T/1 year = [100*(1 - 3/100)]^(3/2)

Further simplifying and approximating, we get:

T/1 year ≈ 100^(3/2) * (1 - 3/100)^(3/2)

Now, let's calculate this expression step-by-step:

(1) Calculate 100^(3/2):

100^(3/2) = 1,000

(2) Calculate (1 - 3/100)^(3/2):

Using a binomial expansion, we can approximate (1 - 3/100)^(3/2) as:

(1 - 3/100)^(3/2) ≈ 1 - (3/2) * (3/100) + (3/2) * (1/2) * (3/100)^2

Simplifying further:

(1 - 3/100)^(3/2) ≈ 1 - 9/200 + (27/8000)

Approximating 27/8000 as 1/300, we can rewrite:

(1 - 3/100)^(3/2) ≈ 1 - 9/200 + 1/300

Combining the expressions, we get:

T/1 year ≈ 1,000 * (1 - 9/200 + 1/300)

Simplifying:

T/1 year ≈ 1,000 - 45 + 1/3

Approximating 1/3 as 0.3333, we get:

T/1 year ≈ 1,000 - 45 + 0.3333

Finally, calculating the sum:

T ≈ 955.3 years

Therefore, it would take approximately 955.3 years for the planet to orbit the sun based on the given information.