Given:
Distance of the planet from the Sun = 5 times the distance between the Earth and the Sun

To find:
Time period of the planet

Solution:
The time period of a planet is the time it takes to complete one revolution around the Sun. It is given by Kepler's Third Law of Planetary Motion, which states that the square of the time period of a planet is proportional to the cube of its average distance from the Sun.

Let the distance between the Earth and the Sun be 'd'.
Distance of the planet from the Sun = 5d

Step 1:
Let's consider the time period of the Earth around the Sun as 'T'. According to Kepler's Third Law, we have the following equation:
T^2 ∝ d^3

Step 2:
Now, let's consider the time period of the planet around the Sun as 'T_p'. According to Kepler's Third Law, we have the following equation:
T_p^2 ∝ (5d)^3

Step 3:
Since both equations are proportional to the cube of the distance, we can equate them:
T^2 = (5d)^3

Step 4:
Simplifying the equation:
T^2 = 125d^3

Step 5:
Dividing both sides of the equation by d^3:
(T^2)/(d^3) = 125

Step 6:
Substituting the value of T^2 from the first equation:
(1^2)/(d^3) = 125

Step 7:
Simplifying the equation:
1/(d^3) = 125

Step 8:
Taking the cube root of both sides of the equation:
1/d = 5

Step 9:
Simplifying the equation:
d = 1/5

Step 10:
Substituting the value of d in the first equation:
T^2 = (1/5)^3

Step 11:
Simplifying the equation:
T^2 = 1/125

Step 12:
Taking the square root of both sides of the equation:
T = 1/√125

Step 13:
Simplifying the equation:
T = 1/5√5

Step 14:
Rationalizing the denominator:
T = 1/(5√5) * (√5/√5)

Step 15:
Simplifying the equation:
T = √5/25

Step 16:
Rearranging the equation:
T = (√5)/5^2

Step 17:
Simplifying the equation:
T = (√5)/25

Step 18:
Converting the square root into a fraction:
T = (√5)/25 * (√5/√5)

Step 19:
Simplifying the equation:
T = 5/25

Step 20:
Simplifying the equation:
T =