If the Geometrical mean of x, 16, 50 is 20, then the value of x is

Options

A. 40
B. 20
C. 10
D. 4

Answer

To find the value of \( x \) given that the geometric mean of \( x, 16, \) and \( 50 \) is \( 20 \), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the formula for Geometric Mean**:
The geometric mean (GM) of \( n \) numbers \( a_1, a_2, \ldots, a_n \) is given by the formula:
\[ GM = \sqrt[n]{a_1 \times a_2 \times \ldots \times a_n} \]
In this case, we have three numbers: \( x, 16, \) and \( 50 \).

2. **Set up the equation**:
According to the problem, the geometric mean of \( x, 16, \) and \( 50 \) is \( 20 \). Therefore, we can write:
\[ \sqrt[3]{x \times 16 \times 50} = 20 \]

3. **Cube both sides**:
To eliminate the cube root, we cube both sides of the equation:
\[ x \times 16 \times 50 = 20^3 \]

4. **Calculate \( 20^3 \)**:
Now, calculate \( 20^3 \):
\[ 20^3 = 20 \times 20 \times 20 = 8000 \]
So, we have:
\[ x \times 16 \times 50 = 8000 \]

5. **Calculate \( 16 \times 50 \)**:
Next, calculate \( 16 \times 50 \):
\[ 16 \times 50 = 800 \]
Thus, we can rewrite the equation as:
\[ x \times 800 = 8000 \]

6. **Solve for \( x \)**:
To find \( x \), divide both sides by \( 800 \):
\[ x = \frac{8000}{800} \]
Simplifying this gives:
\[ x = 10 \]

### Final Answer:
The value of \( x \) is \( 10 \).

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