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 find the values of \( a \) and \( b \) given the mean and variance of the numbers \( a, b, 8, 5, \) and \( 10 \).

### Step 1: Calculate the Mean
The mean of the numbers \( a, b, 8, 5, \) and \( 10 \) is given as \( 6 \).

The formula for the mean is:
\[ \text{Mean} = \frac{a + b + 8 + 5 + 10}{5} \]
Substituting the values:
\[ 6 = \frac{a + b + 23}{5} \]
Multiplying both sides by \( 5 \):
\[ 30 = a + b + 23 \]
Rearranging gives:
\[ a + b = 30 - 23 = 7 \quad \text{(Equation 1)} \]

### Step 2: Calculate the Variance
The variance is given as \( 6.8 \). The formula for variance is:
\[ \text{Variance} = \frac{(a - \text{mean})^2 + (b - \text{mean})^2 + (8 - \text{mean})^2 + (5 - \text{mean})^2 + (10 - \text{mean})^2}{5} \]
Substituting the mean:
\[ 6.8 = \frac{(a - 6)^2 + (b - 6)^2 + (8 - 6)^2 + (5 - 6)^2 + (10 - 6)^2}{5} \]
Calculating the squares:
\[ (8 - 6)^2 = 4, \quad (5 - 6)^2 = 1, \quad (10 - 6)^2 = 16 \]
So the equation becomes:
\[ 6.8 = \frac{(a - 6)^2 + (b - 6)^2 + 4 + 1 + 16}{5} \]
Simplifying:
\[ 6.8 = \frac{(a - 6)^2 + (b - 6)^2 + 21}{5} \]
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 Equation 1 into Equation 2
From Equation 1, we have \( b = 7 - a \). Substitute this into Equation 2:
\[ (a - 6)^2 + ((7 - a) - 6)^2 = 13 \]
This simplifies to:
\[ (a - 6)^2 + (1 - a)^2 = 13 \]
Expanding both squares:
\[ (a^2 - 12a + 36) + (1 - 2a + a^2) = 13 \]
Combining like terms:
\[ 2a^2 - 14a + 37 = 13 \]
Rearranging gives:
\[ 2a^2 - 14a + 24 = 0 \]
Dividing through by \( 2 \):
\[ a^2 - 7a + 12 = 0 \]

### Step 4: Factor the Quadratic
Factoring the quadratic:
\[ (a - 3)(a - 4) = 0 \]
Thus, \( a = 3 \) or \( a = 4 \).

### Step 5: Find Corresponding \( b \)
If \( a = 3 \):
\[ b = 7 - 3 = 4 \]
If \( a = 4 \):
\[ b = 7 - 4 = 3 \]
So, we have \( (a, b) = (3, 4) \) or \( (4, 3) \).

### Step 6: Calculate \( a^3 + b^3 \)
Now we need to calculate \( a^3 + b^3 \):
\[ a^3 + b^3 = 3^3 + 4^3 = 27 + 64 = 91 \]

### Final Answer
Thus, the value of \( a^3 + b^3 \) is \( \boxed{91} \).