If the mean and the variance of the numbers a, b, 8, 5 and 10 are 6 and 6.8 respectively, then the value of `a^(3)+b^(3)` is equal to

Options

A. 58
B. 61
C. 91
D. 89

Answer

To solve the problem, we need to determine the values of \( a \) and \( b \) based on the given mean and variance of the numbers \( a, b, 8, 5, \) and \( 10 \).

### Step 1: Calculate the Mean
The mean of the numbers is given as 6. The formula for the mean is:

\[ \text{Mean} = \frac{a + b + 8 + 5 + 10}{5} \]

Substituting the known values:

\[ 6 = \frac{a + b + 23}{5} \]

Multiplying both sides by 5:

\[ 30 = a + b + 23 \]

Rearranging gives us:

\[ a + b = 30 - 23 = 7 \quad \text{(Equation 1)} \]

### Step 2: Calculate the Variance
The variance is given as 6.8. The formula for variance is:

\[ \text{Variance} = \frac{\sum (x_i - \mu)^2}{n} \]

Where \( \mu \) is the mean, and \( n \) is the number of terms. Substituting the known values:

\[ 6.8 = \frac{(a - 6)^2 + (b - 6)^2 + (8 - 6)^2 + (5 - 6)^2 + (10 - 6)^2}{5} \]

Calculating the known terms:

\[ (8 - 6)^2 = 2^2 = 4 \]
\[ (5 - 6)^2 = (-1)^2 = 1 \]
\[ (10 - 6)^2 = 4^2 = 16 \]

Now substituting back into the variance equation:

\[ 6.8 = \frac{(a - 6)^2 + (b - 6)^2 + 4 + 1 + 16}{5} \]

This simplifies to:

\[ 6.8 = \frac{(a - 6)^2 + (b - 6)^2 + 21}{5} \]

Multiplying both sides by 5:

\[ 34 = (a - 6)^2 + (b - 6)^2 + 21 \]

Rearranging gives:

\[ (a - 6)^2 + (b - 6)^2 = 34 - 21 = 13 \quad \text{(Equation 2)} \]

### Step 3: Substitute \( a \) in terms of \( b \)
From Equation 1, we have \( a = 7 - b \). Substitute this into Equation 2:

\[ ((7 - b) - 6)^2 + (b - 6)^2 = 13 \]

This simplifies to:

\[ (1 - b)^2 + (b - 6)^2 = 13 \]

Expanding both squares:

\[ (1 - 2b + b^2) + (b^2 - 12b + 36) = 13 \]

Combining like terms:

\[ 2b^2 - 14b + 37 = 13 \]

Rearranging gives:

\[ 2b^2 - 14b + 24 = 0 \]

### Step 4: Solve the Quadratic Equation
Dividing the entire equation by 2:

\[ b^2 - 7b + 12 = 0 \]

Factoring gives:

\[ (b - 3)(b - 4) = 0 \]

Thus, the possible values for \( b \) are:

\[ b = 3 \quad \text{or} \quad b = 4 \]

### Step 5: Find Corresponding Values of \( a \)
Using \( a + b = 7 \):

1. If \( b = 3 \), then \( a = 7 - 3 = 4 \).
2. If \( b = 4 \), then \( a = 7 - 4 = 3 \).

### Step 6: Calculate \( a^3 + b^3 \)
Using the identity \( a^3 + b^3 = (a + b)(a^2 - ab + b^2) \):

1. \( a + b = 7 \)
2. \( ab = 3 \times 4 = 12 \) (or \( 4 \times 3 = 12 \))
3. \( a^2 + b^2 = (a + b)^2 - 2ab = 7^2 - 2 \times 12 = 49 - 24 = 25 \)

Now substituting into the identity:

\[ a^3 + b^3 = 7 \times (25 - 12) = 7 \times 13 = 91 \]

### Final Answer
Thus, the value of \( a^3 + b^3 \) is:

\[ \boxed{91} \]