1. A pit in the shape of a rectangular prism has been dug. Its depth is 12 times greater than its length. The length of the pit is $\frac{3}{2}$ times its width. The sum of the numerical value of the volume of the pit and the area of its base is $\frac{7}{6}$. Calculate the depth of the pit.
1. We choose variables $a$ is length, $b$ is width, and $c$ is depth. We set up the system of equations $12 a=c, a=\frac{3 b}{2}$, and $a b c+a b=\frac{7}{6}$. We rearrange the system and get $48 a^{3}+4 a^{2}-7=0$. We solve the cubic equation using Horner's algorithm. We write the solutions $a=\frac{1}{2}, b=\frac{1}{3}$, and $c=6$.

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