Let OABC be a regular tetrahedron of edge length unity. Its volume be V and `6V =sqrt(p/q)` where p and q are relatively prime. The find the value of `(p+q)`.

Answer

To solve the problem, we need to find the volume of a regular tetrahedron with edge length unity and express it in the form given in the question. Here’s a step-by-step solution:

### Step 1: Understand the properties of a regular tetrahedron
A regular tetrahedron is a three-dimensional shape with four triangular faces, all of which are equilateral triangles. The volume \( V \) of a regular tetrahedron with edge length \( a \) is given by the formula:
\[ V = \frac{a^3}{6\sqrt{2}} \]

### Step 2: Substitute the edge length
Since the edge length \( a \) is given as unity (1 unit), we substitute \( a = 1 \) into the volume formula:
\[ V = \frac{1^3}{6\sqrt{2}} = \frac{1}{6\sqrt{2}} \]

### Step 3: Calculate \( 6V \)
Next, we need to calculate \( 6V \):
\[ 6V = 6 \times \frac{1}{6\sqrt{2}} = \frac{1}{\sqrt{2}} \]

### Step 4: Express \( 6V \) in the form \( \sqrt{\frac{p}{q}} \)
We know that:
\[ 6V = \frac{1}{\sqrt{2}} = \sqrt{\frac{1}{2}} \]
This implies that:
\[ \sqrt{\frac{p}{q}} = \sqrt{\frac{1}{2}} \]
From this, we can equate:
\[ \frac{p}{q} = \frac{1}{2} \]

### Step 5: Identify \( p \) and \( q \)
From the equation \( \frac{p}{q} = \frac{1}{2} \), we can see that \( p = 1 \) and \( q = 2 \). Since 1 and 2 are relatively prime (they have no common factors other than 1), this condition is satisfied.

### Step 6: Calculate \( p + q \)
Finally, we need to find the value of \( p + q \):
\[ p + q = 1 + 2 = 3 \]

### Conclusion
Thus, the required value of \( p + q \) is:
\[ \boxed{3} \]

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