The length of two opposite edges of a tetrahedron are 12 and 15 units and the shortest distance between them is 10 units. If the volume of the tetrahedron is 200 cubic units, then the angle between the 2 edges is

Options

A. `sin^(-1).(1)/(2)`
B. `sin^(-1).(2)/(3)`
C. `sin^(-1).(3)/(4)`
D. `sin^(-1).(4)/(5)`

Answer

To find the angle between the two edges of the tetrahedron, we can follow these steps:

### Step 1: Understand the Problem
We are given two opposite edges of a tetrahedron with lengths 12 units and 15 units, the shortest distance between them is 10 units, and the volume of the tetrahedron is 200 cubic units. We need to find the angle between these two edges.

### Step 2: Define the Vectors
Let the two opposite edges be represented by vectors:
- Let \( \vec{AB} \) be the vector representing the edge of length 12 units.
- Let \( \vec{CD} \) be the vector representing the edge of length 15 units.

### Step 3: Use the Shortest Distance Formula
The shortest distance \( d \) between two skew lines (edges) can be calculated using the formula:
\[ d = \frac{|\vec{D} - \vec{B} \cdot (\vec{B} - \vec{A}) \times (\vec{D} - \vec{C})|}{|\vec{B} - \vec{A}| \cdot |\vec{D} - \vec{C}|} \]
Given that \( d = 10 \) units, we can set up the equation:
\[ 10 = \frac{|\vec{D} - \vec{B} \cdot (\vec{B} - \vec{A}) \times (\vec{D} - \vec{C})|}{|\vec{B} - \vec{A}| \cdot |\vec{D} - \vec{C}|} \]

### Step 4: Calculate the Volume of the Tetrahedron
The volume \( V \) of a tetrahedron can be calculated using the formula:
\[ V = \frac{1}{6} |\vec{D} - \vec{A} \cdot ((\vec{C} - \vec{A}) \times (\vec{B} - \vec{A}))| \]
Given that \( V = 200 \) cubic units, we can set up the equation:
\[ 200 = \frac{1}{6} |\vec{D} - \vec{A} \cdot ((\vec{C} - \vec{A}) \times (\vec{B} - \vec{A}))| \]

### Step 5: Relate the Two Equations
From the two equations, we can express the dot product in terms of the sine of the angle \( \theta \) between the two edges:
\[ |\vec{B} - \vec{A}| = 12, \quad |\vec{D} - \vec{C}| = 15 \]
Thus, we can rewrite the volume equation:
\[ |\vec{D} - \vec{B} \cdot (\vec{B} - \vec{A}) \times (\vec{D} - \vec{C})| = 120 \]
This can be expressed as:
\[ |\vec{B} - \vec{A}| \cdot |\vec{D} - \vec{C}| \cdot \sin(\theta) = 120 \]

### Step 6: Substitute Values
Substituting the lengths of the edges:
\[ 12 \cdot 15 \cdot \sin(\theta) = 120 \]
This simplifies to:
\[ 180 \sin(\theta) = 120 \]
Thus,
\[ \sin(\theta) = \frac{120}{180} = \frac{2}{3} \]

### Step 7: Find the Angle
To find the angle \( \theta \), we take the inverse sine:
\[ \theta = \sin^{-1}\left(\frac{2}{3}\right) \]

### Final Answer
The angle between the two edges of the tetrahedron is:
\[ \theta = \sin^{-1}\left(\frac{2}{3}\right) \]