If `h(x) = sqrt(x^(2)+x-4)` and `g(x)=sqrt(5x)`, what is the value of `h(g(6))` ?

Options

A. `5.48`
B. `5.55`
C. `5.61`
D. `5.83`

Explanation

To find the value of \( h(g(6)) \) where \( h(x) = \sqrt{x^2 + x - 4} \) and \( g(x) = \sqrt{5x} \), we will follow these steps:

### Step 1: Calculate \( g(6) \)
We start by substituting \( x = 6 \) into the function \( g(x) \).

\[ g(6) = \sqrt{5 \cdot 6} = \sqrt{30} \]

### Step 2: Substitute \( g(6) \) into \( h(x) \)
Now, we need to find \( h(g(6)) \), which is \( h(\sqrt{30}) \).

\[ h(\sqrt{30}) = \sqrt{(\sqrt{30})^2 + \sqrt{30} - 4} \]

### Step 3: Simplify the expression inside \( h \)
Calculating \( (\sqrt{30})^2 \):

\[ (\sqrt{30})^2 = 30 \]

Now substituting this back into the expression for \( h(\sqrt{30}) \):

\[ h(\sqrt{30}) = \sqrt{30 + \sqrt{30} - 4} \]

### Step 4: Combine the terms
Now, simplify the expression:

\[ h(\sqrt{30}) = \sqrt{30 - 4 + \sqrt{30}} = \sqrt{26 + \sqrt{30}} \]

### Step 5: Calculate \( \sqrt{30} \) and then \( h(g(6)) \)
Using a calculator, we find \( \sqrt{30} \approx 5.477 \). Now substituting this value back into the expression:

\[ h(g(6)) = \sqrt{26 + 5.477} = \sqrt{31.477} \]

### Step 6: Calculate \( \sqrt{31.477} \)
Using a calculator again, we find:

\[ \sqrt{31.477} \approx 5.61 \]

Thus, the value of \( h(g(6)) \) is approximately \( 5.61 \).

### Final Answer
\[ h(g(6)) \approx 5.61 \]