. Let a, b > 0 and `vecalpha=hati/a+4hatj/b+bhatk` and `beta=bhati+ahatj+hatk/b` then the maximum value of `30/(5+alpha.beta)`

Options

A. `3`
B. `2`
C. `4`
D. `8`

Answer

To find the maximum value of the expression \( \frac{30}{5 + \vec{\alpha} \cdot \vec{\beta}} \), we will first compute the dot product \( \vec{\alpha} \cdot \vec{\beta} \) and then analyze its minimum value.

### Step 1: Define the vectors
The vectors are given as:
\[ \vec{\alpha} = \frac{\hat{i}}{a} + \frac{4\hat{j}}{b} + b\hat{k} \]
\[ \vec{\beta} = b\hat{i} + a\hat{j} + \frac{\hat{k}}{b} \]

### Step 2: Compute the dot product \( \vec{\alpha} \cdot \vec{\beta} \)
The dot product \( \vec{\alpha} \cdot \vec{\beta} \) can be calculated as follows:
\[ \vec{\alpha} \cdot \vec{\beta} = \left(\frac{\hat{i}}{a} \cdot b\hat{i}\right) + \left(\frac{4\hat{j}}{b} \cdot a\hat{j}\right) + \left(b\hat{k} \cdot \frac{\hat{k}}{b}\right) \]
Calculating each term:
1. \( \frac{\hat{i}}{a} \cdot b\hat{i} = \frac{b}{a} \)
2. \( \frac{4\hat{j}}{b} \cdot a\hat{j} = \frac{4a}{b} \)
3. \( b\hat{k} \cdot \frac{\hat{k}}{b} = 1 \)

Combining these, we have:
\[ \vec{\alpha} \cdot \vec{\beta} = \frac{b}{a} + \frac{4a}{b} + 1 \]

### Step 3: Find the minimum value of \( \vec{\alpha} \cdot \vec{\beta} \)
To find the minimum value of \( \frac{b}{a} + \frac{4a}{b} + 1 \), we can apply the AM-GM inequality:
\[ \frac{\frac{b}{a} + \frac{4a}{b}}{2} \geq \sqrt{\frac{b}{a} \cdot \frac{4a}{b}} = \sqrt{4} = 2 \]
Thus,
\[ \frac{b}{a} + \frac{4a}{b} \geq 4 \]
Adding 1 to both sides gives:
\[ \frac{b}{a} + \frac{4a}{b} + 1 \geq 5 \]
The minimum value of \( \vec{\alpha} \cdot \vec{\beta} \) is therefore 5.

### Step 4: Substitute into the original expression
Now substituting this minimum value into the expression \( \frac{30}{5 + \vec{\alpha} \cdot \vec{\beta}} \):
\[ \frac{30}{5 + \vec{\alpha} \cdot \vec{\beta}} \leq \frac{30}{5 + 5} = \frac{30}{10} = 3 \]

### Conclusion
Thus, the maximum value of \( \frac{30}{5 + \vec{\alpha} \cdot \vec{\beta}} \) is:
\[ \boxed{3} \]