If the standard deviation of the numbers 2, 4, a and 10 is 3.5, then which of the following is true?

Options

A. `3a^(2)-23a+24=0`
B. `3a^(2)-26a+46=0`
C. `3a^(2)-32a+28=0`
D. `3a^(2)-34a+45=0`

Answer

To solve the problem, we need to find the value of \( a \) such that the standard deviation of the numbers \( 2, 4, a, \) and \( 10 \) is \( 3.5 \).

### Step-by-Step Solution:

1. **Understand the Standard Deviation Formula**:
The standard deviation \( \sigma \) is given by the formula:
\[ \sigma = \sqrt{\frac{\sum (x_i - \mu)^2}{n}} \]
where \( \mu \) is the mean of the numbers, and \( n \) is the number of values.

2. **Calculate the Mean**:
The mean \( \mu \) of the numbers \( 2, 4, a, 10 \) is:
\[ \mu = \frac{2 + 4 + a + 10}{4} = \frac{16 + a}{4} \]

3. **Calculate the Variance**:
The variance \( \sigma^2 \) is given by:
\[ \sigma^2 = \frac{(2 - \mu)^2 + (4 - \mu)^2 + (a - \mu)^2 + (10 - \mu)^2}{4} \]
We know that \( \sigma = 3.5 \), so:
\[ \sigma^2 = (3.5)^2 = 12.25 \]

4. **Substituting the Mean into the Variance Formula**:
We need to compute each term:
- \( (2 - \mu)^2 = \left(2 - \frac{16 + a}{4}\right)^2 = \left(\frac{8 - a}{4}\right)^2 = \frac{(8 - a)^2}{16} \)
- \( (4 - \mu)^2 = \left(4 - \frac{16 + a}{4}\right)^2 = \left(\frac{16 - a}{4}\right)^2 = \frac{(16 - a)^2}{16} \)
- \( (a - \mu)^2 = \left(a - \frac{16 + a}{4}\right)^2 = \left(\frac{3a - 16}{4}\right)^2 = \frac{(3a - 16)^2}{16} \)
- \( (10 - \mu)^2 = \left(10 - \frac{16 + a}{4}\right)^2 = \left(\frac{24 - a}{4}\right)^2 = \frac{(24 - a)^2}{16} \)

5. **Combine the Variance Terms**:
The variance becomes:
\[ \sigma^2 = \frac{\frac{(8 - a)^2 + (16 - a)^2 + (3a - 16)^2 + (24 - a)^2}{16}}{4} \]
Simplifying this gives:
\[ \sigma^2 = \frac{(8 - a)^2 + (16 - a)^2 + (3a - 16)^2 + (24 - a)^2}{64} \]
Setting this equal to \( 12.25 \):
\[ \frac{(8 - a)^2 + (16 - a)^2 + (3a - 16)^2 + (24 - a)^2}{64} = 12.25 \]
Multiplying both sides by \( 64 \):
\[ (8 - a)^2 + (16 - a)^2 + (3a - 16)^2 + (24 - a)^2 = 784 \]

6. **Expand and Simplify**:
Expanding each term:
- \( (8 - a)^2 = 64 - 16a + a^2 \)
- \( (16 - a)^2 = 256 - 32a + a^2 \)
- \( (3a - 16)^2 = 9a^2 - 96a + 256 \)
- \( (24 - a)^2 = 576 - 48a + a^2 \)

Combining these gives:
\[ 64 + 256 + 576 + (1 + 1 + 9 + 1)a^2 - (16 + 32 + 96 + 48)a = 784 \]
\[ 896 + 12a^2 - 192a = 784 \]
Rearranging gives:
\[ 12a^2 - 192a + 112 = 0 \]

7. **Solve the Quadratic Equation**:
Dividing through by 4:
\[ 3a^2 - 48a + 28 = 0 \]
Using the quadratic formula:
\[ a = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{48 \pm \sqrt{(-48)^2 - 4 \cdot 3 \cdot 28}}{2 \cdot 3} \]
\[ a = \frac{48 \pm \sqrt{2304 - 336}}{6} = \frac{48 \pm \sqrt{1968}}{6} \]
Simplifying gives:
\[ a = \frac{48 \pm 14\sqrt{2}}{6} \]

8. **Determine the Valid Options**:
Based on the calculations, we can check which of the options provided is valid.