Let `p`,`q`, `r in R^(+)` and `27pqr ge (p+q+r)^(3)` and `3p+4q+5r=12`. Then the value of `8p+4q-7r=`

Options

A. `2`
B. `3`
C. `4`
D. `5`

Explanation

To solve the problem, we will follow these steps:

### Step 1: Understand the Given Inequality
We are given that \( 27pqr \geq (p + q + r)^3 \). This is a form of the Arithmetic Mean-Geometric Mean (AM-GM) inequality.

### Step 2: Apply AM-GM Inequality
By the AM-GM inequality, we know that:
\[ \frac{p + q + r}{3} \geq \sqrt[3]{pqr} \]
Cubing both sides gives us:
\[ \left(\frac{p + q + r}{3}\right)^3 \geq pqr \]
Multiplying both sides by 27 results in:
\[ (p + q + r)^3 \geq 27pqr \]
This tells us that equality holds when \( p = q = r \).

### Step 3: Set \( p = q = r \)
Let \( p = q = r = k \). Then we can substitute this into the equation \( 3p + 4q + 5r = 12 \).

### Step 4: Substitute into the Equation
Substituting \( p = k \), \( q = k \), and \( r = k \) into the equation:
\[ 3k + 4k + 5k = 12 \]
This simplifies to:
\[ 12k = 12 \]
Thus, we find:
\[ k = 1 \]

### Step 5: Find Values of \( p, q, r \)
Since \( p = q = r = k \), we have:
\[ p = 1, \quad q = 1, \quad r = 1 \]

### Step 6: Calculate \( 8p + 4q - 7r \)
Now we substitute \( p, q, r \) into the expression \( 8p + 4q - 7r \):
\[ 8p + 4q - 7r = 8(1) + 4(1) - 7(1) \]
This simplifies to:
\[ 8 + 4 - 7 = 5 \]

### Final Answer
Thus, the value of \( 8p + 4q - 7r \) is \( \boxed{5} \).