A 6 in thick slice is cut off the top of a cube, resulting in a rectangular box that has volume $129 \mathrm{in}^{3}$. Use the ALEKS graphing calculator to find the side length of the original cube. Round your answer to two decimal places.
Answer
Finally, we check that this solution makes sense in the context of the problem. Since the slice cut off the cube was 6 inches thick, the side length of the cube must be greater than 6 inches. Our solution of 7.68 inches satisfies this requirement.
Step 1 ：Let's denote the side length of the original cube as \(s\). The volume of the cube is \(s^3\).
Step 2 ：When a 6 in thick slice is cut off the top of the cube, the height of the resulting rectangular box is \(s-6\). The length and width remain \(s\). Therefore, the volume of the rectangular box is \(s \cdot s \cdot (s-6) = s^3 - 6s^2\).
Step 3 ：We know that the volume of the rectangular box is 129 cubic inches, so we can set up the equation \(s^3 - 6s^2 = 129\).
Step 4 ：Rearranging the equation gives \(s^3 - 6s^2 - 129 = 0\).
Step 5 ：This is a cubic equation, which can be solved using methods such as factoring, the rational root theorem, or numerical methods. In this case, we are asked to use the ALEKS graphing calculator to find the solution.
Step 6 ：By graphing the function \(f(s) = s^3 - 6s^2 - 129\) and finding the root, we get the side length of the original cube, \(s\), to be approximately 7.68 inches when rounded to two decimal places.
Step 7 ：Finally, we check that this solution makes sense in the context of the problem. Since the slice cut off the cube was 6 inches thick, the side length of the cube must be greater than 6 inches. Our solution of 7.68 inches satisfies this requirement.
