Two satelites of a planet have period 32 days and 256 days. If the radius of orbit of former is R, find the orbital radius of the latter.

Explanation

To solve the problem of finding the orbital radius of the second satellite given the period of both satellites, we can use Kepler's Third Law of Planetary Motion, which states that the square of the period of a satellite is directly proportional to the cube of the semi-major axis of its orbit.

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Period of the first satellite (T1) = 32 days
- Period of the second satellite (T2) = 256 days
- Radius of the first satellite's orbit (R1) = R (unknown value)
- Radius of the second satellite's orbit (R2) = ? (to be found)

2. **Apply Kepler's Third Law:**
According to Kepler's Third Law:
\[ \frac{T_1^2}{T_2^2} = \frac{R_1^3}{R_2^3} \]

3. **Substitute the Known Values:**
Substitute T1 and T2 into the equation:
\[ \frac{(32)^2}{(256)^2} = \frac{R^3}{R_2^3} \]

4. **Calculate the Left Side:**
Calculate \(32^2\) and \(256^2\):
\[ 32^2 = 1024 \]
\[ 256^2 = 65536 \]
Therefore:
\[ \frac{1024}{65536} = \frac{R^3}{R_2^3} \]

5. **Simplify the Fraction:**
Simplifying \( \frac{1024}{65536} \):
\[ \frac{1024}{65536} = \frac{1}{64} \]
So we have:
\[ \frac{1}{64} = \frac{R^3}{R_2^3} \]

6. **Cross Multiply:**
Cross multiplying gives:
\[ R_2^3 = 64R^3 \]

7. **Take the Cube Root:**
Taking the cube root of both sides:
\[ R_2 = R \cdot 4 \]

8. **Final Answer:**
The orbital radius of the second satellite is:
\[ R_2 = 4R \]

### Summary:
The orbital radius of the second satellite is four times the radius of the first satellite.