Let the function g be defined by `g(x)=5x+2`. If `sqrt(g((a)/(2)))=6`, what is the value of a?

Options

A. `(1)/(sqrt(6))`
B. `(1)/(sqrt(2))`
C. `(34)/(5)`
D. `(68)/(5)`

Answer

To solve the problem step by step, we will follow the given function and the equation provided.

### Step 1: Understand the function
The function is defined as:
\[ g(x) = 5x + 2 \]

### Step 2: Substitute \( x \) with \( \frac{a}{2} \)
We need to find \( g\left(\frac{a}{2}\right) \):
\[ g\left(\frac{a}{2}\right) = 5\left(\frac{a}{2}\right) + 2 \]

### Step 3: Simplify the expression
Now simplify the expression:
\[ g\left(\frac{a}{2}\right) = \frac{5a}{2} + 2 \]

### Step 4: Set up the equation with the given condition
According to the problem, we have:
\[ \sqrt{g\left(\frac{a}{2}\right)} = 6 \]
Substituting our expression from Step 3:
\[ \sqrt{\frac{5a}{2} + 2} = 6 \]

### Step 5: Square both sides to eliminate the square root
Now, square both sides of the equation:
\[ \frac{5a}{2} + 2 = 6^2 \]
\[ \frac{5a}{2} + 2 = 36 \]

### Step 6: Isolate the term with \( a \)
Subtract 2 from both sides:
\[ \frac{5a}{2} = 36 - 2 \]
\[ \frac{5a}{2} = 34 \]

### Step 7: Solve for \( a \)
Now, multiply both sides by 2 to eliminate the fraction:
\[ 5a = 34 \times 2 \]
\[ 5a = 68 \]

Now, divide both sides by 5:
\[ a = \frac{68}{5} \]

### Final Answer
Thus, the value of \( a \) is:
\[ a = \frac{68}{5} \]

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