If `Sigma_(i=1)^(10) x_i=60` and `Sigma_(i=1)^(10)x_i^2=360` then `Sigma_(i=1)^(10)x_i^3` is

Options

A. 2160
B. 3250
C. 3360
D. None of these

Answer

To solve the problem, we need to find the value of \( \Sigma_{i=1}^{10} x_i^3 \) given that \( \Sigma_{i=1}^{10} x_i = 60 \) and \( \Sigma_{i=1}^{10} x_i^2 = 360 \).

### Step-by-Step Solution:

1. **Identify the Given Information:**
- We have \( \Sigma_{i=1}^{10} x_i = 60 \) (sum of the observations).
- We also have \( \Sigma_{i=1}^{10} x_i^2 = 360 \) (sum of the squares of the observations).

2. **Calculate the Mean (\( \bar{x} \)):**
- The mean \( \bar{x} \) can be calculated as:
\[ \bar{x} = \frac{\Sigma_{i=1}^{10} x_i}{10} = \frac{60}{10} = 6 \]

3. **Calculate the Variance:**
- The variance \( \sigma^2 \) is given by the formula:
\[ \sigma^2 = \frac{\Sigma_{i=1}^{10} x_i^2}{n} - \bar{x}^2 \]
- Substituting the known values:
\[ \sigma^2 = \frac{360}{10} - 6^2 = 36 - 36 = 0 \]

4. **Interpret the Variance:**
- A variance of 0 indicates that all observations are the same. Therefore, we can conclude:
\[ x_1 = x_2 = x_3 = \ldots = x_{10} = x \]

5. **Find the Value of \( x \):**
- Since \( \Sigma_{i=1}^{10} x_i = 60 \):
\[ 10x = 60 \implies x = 6 \]

6. **Calculate \( \Sigma_{i=1}^{10} x_i^3 \):**
- Since all \( x_i \) are equal to 6:
\[ \Sigma_{i=1}^{10} x_i^3 = 10 \cdot (6^3) = 10 \cdot 216 = 2160 \]

### Final Answer:
\[ \Sigma_{i=1}^{10} x_i^3 = 2160 \]