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. a) `2160`
B. b) `3250`
C. c) `3360`
D. 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. **Understanding the Given Information**:
We have two important equations:
\[ \Sigma_{i=1}^{10} x_i = 60 \]
\[ \Sigma_{i=1}^{10} x_i^2 = 360 \]

2. **Using the Variance Formula**:
The variance \( \sigma^2 \) can be calculated using the formula:
\[ \sigma^2 = \frac{\Sigma x_i^2}{n} - \left(\frac{\Sigma x_i}{n}\right)^2 \]
where \( n \) is the number of observations. Here, \( n = 10 \).

3. **Substituting the Values**:
Substitute the known values into the variance formula:
\[ \sigma^2 = \frac{360}{10} - \left(\frac{60}{10}\right)^2 \]
Simplifying this gives:
\[ \sigma^2 = 36 - 6^2 = 36 - 36 = 0 \]

4. **Interpreting the Variance**:
Since the variance is 0, it implies that all the values \( x_i \) are the same. Let’s denote this common value as \( x \). Therefore:
\[ x_1 = x_2 = x_3 = \ldots = x_{10} = x \]

5. **Finding the Value of \( x \)**:
From the first equation \( \Sigma_{i=1}^{10} x_i = 60 \):
\[ 10x = 60 \]
Solving for \( x \):
\[ x = \frac{60}{10} = 6 \]

6. **Calculating \( \Sigma_{i=1}^{10} x_i^3 \)**:
Now, we need to find \( \Sigma_{i=1}^{10} x_i^3 \):
\[ \Sigma_{i=1}^{10} x_i^3 = x^3 + x^3 + \ldots + x^3 \quad (10 \text{ times}) = 10x^3 \]
Substituting \( x = 6 \):
\[ \Sigma_{i=1}^{10} x_i^3 = 10 \times 6^3 = 10 \times 216 = 2160 \]

7. **Final Answer**:
Therefore, the value of \( \Sigma_{i=1}^{10} x_i^3 \) is:
\[ \boxed{2160} \]