Suppose `a,b,c epsilon R` and
`Delta=|(a,a+b,a+b+c),(3a,4a+3b,5a+4b+3c),(6a,9a+6b,11a+9b+6c)|` If `Delta=11.728` then a=______

Answer

To solve the problem, we need to evaluate the determinant given by:

\[ \Delta = \begin{vmatrix} a & a+b & a+b+c \\ 3a & 4a+3b & 5a+4b+3c \\ 6a & 9a+6b & 11a+9b+6c \end{vmatrix} \]

We are given that \(\Delta = 11.728\).

### Step 1: Apply properties of determinants

We can simplify the determinant by using the properties of determinants. We will replace the second row \(R_2\) with \(R_2 - 3R_1\) and the third row \(R_3\) with \(R_3 - 6R_1\).

\[ R_2 \rightarrow R_2 - 3R_1 \]
\[ R_3 \rightarrow R_3 - 6R_1 \]

This gives us:

\[ \Delta = \begin{vmatrix} a & a+b & a+b+c \\ 3a - 3a & (4a + 3b) - 3(a + b) & (5a + 4b + 3c) - 3(a + b + c) \\ 6a - 6a & (9a + 6b) - 6(a + b) & (11a + 9b + 6c) - 6(a + b + c) \end{vmatrix} \]

### Step 2: Simplify the determinant

Calculating the new rows:

- For \(R_2\):
- First element: \(3a - 3a = 0\)
- Second element: \(4a + 3b - 3a - 3b = a\)
- Third element: \(5a + 4b + 3c - 3a - 3b - 3c = 2a + b\)

- For \(R_3\):
- First element: \(6a - 6a = 0\)
- Second element: \(9a + 6b - 6a - 6b = 3a\)
- Third element: \(11a + 9b + 6c - 6a - 6b - 6c = 5a + 3b\)

Thus, we have:

\[ \Delta = \begin{vmatrix} a & a+b & a+b+c \\ 0 & a & 2a + b \\ 0 & 3a & 5a + 3b \end{vmatrix} \]

### Step 3: Expand the determinant

Since the first column has two zeros, we can expand along the first column:

\[ \Delta = a \begin{vmatrix} a & 2a + b \\ 3a & 5a + 3b \end{vmatrix} \]

Calculating the 2x2 determinant:

\[ = a \left( a(5a + 3b) - (2a + b)(3a) \right) \]
\[ = a \left( 5a^2 + 3ab - (6a^2 + 3ab) \right) \]
\[ = a \left( 5a^2 + 3ab - 6a^2 - 3ab \right) \]
\[ = a \left( -a^2 \right) \]
\[ = -a^3 \]

### Step 4: Set the determinant equal to the given value

We know that:

\[ -a^3 = 11.728 \]

Thus:

\[ a^3 = -11.728 \]

### Step 5: Solve for \(a\)

Taking the cube root:

\[ a = -\sqrt[3]{11.728} \]

Calculating the cube root:

\[ a \approx -2.272 \]

### Final Answer

Thus, the value of \(a\) is:

\[ \boxed{-2.272} \]