Find the which of the following are correct using dimensional analysis.
(i) `t^(2)=4pi^(2)a^(3)` (ii) `t^(2)=(4pi^(2)a^(3))/G` (iii) `t^(2)=(4pi^(2)a^(3))/(GM)`
where t is the time period, a is the radius of the orbit of a planet and M is the mass of sun.

Explanation

To solve the problem using dimensional analysis, we will analyze each equation step by step.

### Step 1: Analyze the first equation \( t^2 = 4\pi^2 a^3 \)

1. **Identify dimensions:**
- The left-hand side (LHS) is \( t^2 \), which has the dimension of time squared: \([T^2]\).
- The right-hand side (RHS) is \( 4\pi^2 a^3 \). Here, \( 4\pi^2 \) is a dimensionless constant, and \( a \) is the radius with the dimension of length: \([L]\). Therefore, \( a^3 \) has the dimension of length cubed: \([L^3]\).

2. **Combine dimensions on the RHS:**
- The dimension of the RHS is \([L^3]\).

3. **Compare dimensions:**
- LHS: \([T^2]\)
- RHS: \([L^3]\)

Since \([T^2] \neq [L^3]\), the first equation is **not correct**.

### Step 2: Analyze the second equation \( t^2 = \frac{4\pi^2 a^3}{G} \)

1. **Identify dimensions:**
- LHS: \( t^2 \) has the dimension \([T^2]\).
- RHS: \( \frac{4\pi^2 a^3}{G} \).
- \( 4\pi^2 \) is dimensionless.
- \( a^3 \) has the dimension \([L^3]\).
- The gravitational constant \( G \) has the dimension \([M^{-1} L^3 T^{-2}]\).

2. **Combine dimensions on the RHS:**
- The dimension of \( \frac{a^3}{G} \) can be calculated as follows:
\[ \text{Dimension of } \frac{a^3}{G} = \frac{[L^3]}{[M^{-1} L^3 T^{-2}]} = [M^1 T^2] \]
- Thus, the dimension of the RHS is \([M^1 T^2]\).

3. **Compare dimensions:**
- LHS: \([T^2]\)
- RHS: \([M^1 T^2]\)

Since \([T^2] \neq [M^1 T^2]\), the second equation is **not correct**.

### Step 3: Analyze the third equation \( t^2 = \frac{4\pi^2 a^3}{GM} \)

1. **Identify dimensions:**
- LHS: \( t^2 \) has the dimension \([T^2]\).
- RHS: \( \frac{4\pi^2 a^3}{GM} \).
- \( 4\pi^2 \) is dimensionless.
- \( a^3 \) has the dimension \([L^3]\).
- The gravitational constant \( G \) has the dimension \([M^{-1} L^3 T^{-2}]\).
- The mass \( M \) has the dimension \([M]\).

2. **Combine dimensions on the RHS:**
- The dimension of \( \frac{a^3}{GM} \) can be calculated as follows:
\[ \text{Dimension of } \frac{a^3}{GM} = \frac{[L^3]}{[M^{-1} L^3 T^{-2}] \cdot [M]} = \frac{[L^3]}{[M^{-1} L^3 T^{-2}] \cdot [M]} = [T^2] \]
- Thus, the dimension of the RHS is \([T^2]\).

3. **Compare dimensions:**
- LHS: \([T^2]\)
- RHS: \([T^2]\)

Since \([T^2] = [T^2]\), the third equation is **correct**.

### Summary of Results:
- (i) Not correct
- (ii) Not correct
- (iii) Correct