The corrected equation is:
\[ 13 \times 27 + 2w(13 + 27 + 4w) = 384 \]
Now, let's simplify and solve for \( w \):
\[ 351 + 2w(40 + 4w) = 384 \]
\[ 351 + 80w + 8w^2 = 384 \]
Subtract 384 from both sides:
\[ 8w^2 + 80w - 33 = 0 \]
Now, factor or use the quadratic formula to solve for \( w \). After solving, you should get two potential solutions for \( w \). Choose the positive solution since the width cannot be negative in this context.