Answer
$3\ m$
Work Step by Step
Step 1. Draw a diagram as shown in the figure, where $x$ is the width of the path.
Step 2. The dimensions of the larger rectangle are $15+2x$ by $12+2x$, and the area is given by $(15+2x)(12+2x)=378$
Step 3. Simplify the above equation to get $(15+2x)(6+x)=189$, $2x^2+27x+90=189$ and $2x^2+27x-99=0$
Step 4. Solve the above equation by factoring to get $(x-3)(2x+33)=0$. Discard the negative solution to get $x=3\ m$.
