What is the largest possible area of a quadrilateral with sides $1,4,7,8$ ?

$ \textbf{(A)}\ 7\sqrt 2
\qquad\textbf{(B)}\ 10\sqrt 3
\qquad\textbf{(C)}\ 18
\qquad\textbf{(D)}\ 12\sqrt 3
\qquad\textbf{(E)}\ 9\sqrt 5
$
1. To maximize the area of a quadrilateral with given side lengths, the quadrilateral should be cyclic. This is because a cyclic quadrilateral has the maximum possible area for given side lengths.
2. We use Brahmagupta's formula to find the area of a cyclic quadrilateral. Brahmagupta's formula states that the area \( A \) of a cyclic quadrilateral with sides \( a, b, c, d \) is given by:
   \[
   A = \sqrt{(s-a)(s-b)(s-c)(s-d)}
   \]
   where \( s \) is the semiperimeter of the quadrilateral, defined as:
   \[
   s = \frac{a + b + c + d}{2}
   \]
3. For the given sides \( a = 1 \), \( b = 4 \), \( c = 7 \), and \( d = 8 \), we first calculate the semiperimeter \( s \):
   \[
   s = \frac{1 + 4 + 7 + 8}{2} = \frac{20}{2} = 10
   \]
4. Next, we substitute the values into Brahmagupta's formula:
   \[
   A = \sqrt{(10-1)(10-4)(10-7)(10-8)}
   \]
5. Simplify the expression inside the square root:
   \[
   A = \sqrt{(9)(6)(3)(2)}
   \]
6. Calculate the product inside the square root:
   \[
   9 \times 6 = 54
   \]
   \[
   54 \times 3 = 162
   \]
   \[
   162 \times 2 = 324
   \]
7. Finally, take the square root of 324:
   \[
   A = \sqrt{324} = 18
   \]

The final answer is \(\boxed{18}\)