The geometric mean of a set of observations is computed as 10. The geometric mean obtained when each observation `x_(i)` is replaced by `3x_(i)^(4)` is

Options

A. 810
B. 900
C. 30000
D. 81000

Answer

To solve the problem, we need to find the new geometric mean when each observation \( x_i \) is replaced by \( 3x_i^4 \). We start with the given geometric mean of the original observations.

### Step-by-Step Solution:

1. **Understanding the Geometric Mean (GM)**:
The geometric mean of a set of observations \( x_1, x_2, \ldots, x_n \) is given by:
\[ GM = \sqrt[n]{x_1 \cdot x_2 \cdot \ldots \cdot x_n} \]
We know that this GM equals 10:
\[ \sqrt[n]{x_1 \cdot x_2 \cdot \ldots \cdot x_n} = 10 \]

2. **Replacing Observations**:
Each observation \( x_i \) is replaced by \( 3x_i^4 \). Therefore, the new observations are \( 3x_1^4, 3x_2^4, \ldots, 3x_n^4 \).

3. **Calculating the New GM**:
The new geometric mean (GM') can be calculated as:
\[ GM' = \sqrt[n]{(3x_1^4) \cdot (3x_2^4) \cdots (3x_n^4)} \]

4. **Factoring Out Constants**:
We can factor out the constants:
\[ GM' = \sqrt[n]{3^n \cdot (x_1^4 \cdot x_2^4 \cdots x_n^4)} \]
This simplifies to:
\[ GM' = \sqrt[n]{3^n} \cdot \sqrt[n]{(x_1^4 \cdot x_2^4 \cdots x_n^4)} \]

5. **Simplifying Further**:
The term \( \sqrt[n]{3^n} \) simplifies to \( 3 \), and the product \( x_1^4 \cdot x_2^4 \cdots x_n^4 \) can be rewritten as \( (x_1 \cdot x_2 \cdots x_n)^4 \):
\[ GM' = 3 \cdot \sqrt[n]{(x_1 \cdot x_2 \cdots x_n)^4} \]
This can be expressed as:
\[ GM' = 3 \cdot \left(\sqrt[n]{(x_1 \cdot x_2 \cdots x_n)}\right)^4 \]

6. **Substituting the Known GM**:
We know that \( \sqrt[n]{(x_1 \cdot x_2 \cdots x_n)} = 10 \):
\[ GM' = 3 \cdot (10)^4 \]

7. **Calculating \( 10^4 \)**:
We calculate \( 10^4 = 10000 \):
\[ GM' = 3 \cdot 10000 = 30000 \]

### Final Answer:
The new geometric mean when each observation \( x_i \) is replaced by \( 3x_i^4 \) is:
\[ \boxed{30000} \]