A trapezoidal prism whose cross section is shown across is notched out of timber cube of length 12 cm. If the prism is 12 cm long, find what volume of the timber is left in the cube.

Step 1: Calculate the volume of the timber cube. The side length of the cube is $12$ cm. The formula for the volume of a cube is $V_{cube} = s^3$. $V_{cube} = (12 cm)^3$ $V_{cube} = 1728 cm^3$

Step 2: Calculate the area of the trapezoidal cross-section. The parallel sides of the trapezoid are $a = 8.7$ cm and $b = 6.4$ cm. The height of the trapezoid is $h = 5.8$ cm. The formula for the area of a trapezoid is $A_{trapezoid} = \frac{1}{2}(a+b)h$. $A_{trapezoid} = \frac{1}{2}(8.7 cm + 6.4 cm)(5.8 cm)$ $A_{trapezoid} = \frac{1}{2}(15.1 cm)(5.8 cm)$ $A_{trapezoid} = (15.1 cm)(2.9 cm)$ $A_{trapezoid} = 43.79 cm^2$

Step 3: Calculate the volume of the trapezoidal prism. The length of the prism is $L = 12$ cm. The formula for the volume of a prism is $V_{prism} = A_{base} \times L$. $V_{prism} = (43.79 cm^2)(12 cm)$ $V_{prism} = 525.48 cm^3$

Step 4: Calculate the volume of timber left in the cube. The volume of timber left is the volume of the cube minus the volume of the prism. $V_{left} = V_{cube} - V_{prism}$ $V_{left} = 1728 cm^3 - 525.48 cm^3$ $V_{left} = 1202.52 cm^3$

The volume of the timber left in the cube is $\boxed{1202.52 cm^3}$.