Answer
$3$ meters
Work Step by Step
The dimensions of the lot are $15+2x$ and $12+2x$. We are given that the area of the lot is $378$. We solve the equation $Area=378$ for $x$:
$$\begin{align*}
(15+2x)(12+2x)&=378\quad&&\text{Write the given equation.}\\
180+30x+24x+4x^2-378&=0\quad&&\text{Subtract }378\text{ from each side.}\\
4x^2+54x-198&=0\quad&&\text{Simplify.}\\
2x^2+27x-99&=0\quad&&\text{Divide each term by } 2.\\
(x-3)(2x+33)&=0\quad&&\text{Factor.}\\
x-3=0\text{ or }2x+33&=0\quad&&\text{Set each factor equal to 0.}\\
x=3\text{ or }x&=-16.5\quad&&\text{Solve the resulting equations.}
\end{align*}$$
As $x$ is a dimension, it must be positive: the only solution is $x=3$. So the width of the path is $3$ meters.
