Question

a 8 in thick slice is cut off the top of a cube, resulting in a rectangular box that has volume 98 in³. use the aleks graphing calculator to find the side length of the original cube. round your answer to two decimal places.

Explanation:

Step1: Set up the equation

Let the side length of the original cube be \(x\) inches. After cutting off an \(8\) - inch thick slice, the dimensions of the rectangular box are \(x\times x\times(x - 8)\). The volume formula for a rectangular box is \(V=l\times w\times h\). So, we have the equation \(x\cdot x\cdot(x - 8)=98\), which simplifies to \(x^{3}-8x^{2}-98 = 0\).

Step2: Use a graphing calculator

We can use the ALEKS graphing calculator (or a general - purpose graphing calculator). We graph the function \(y=x^{3}-8x^{2}-98\).
Another way is to use the rational root theorem. The possible rational roots of the equation \(x^{3}-8x^{2}-98 = 0\) are factors of \(98\) (i.e., \(\pm1,\pm2,\pm7,\pm14,\pm49,\pm98\)). By testing these values, we find that they are not roots. So, we rely on the graphing calculator.
When we graph \(y=x^{3}-8x^{2}-98\), we look for the \(x\) - intercept (since \(y = 0\) for the roots of the equation \(x^{3}-8x^{2}-98 = 0\)).
Using the graphing calculator, we find that \(x\approx9.28\) (rounded to two decimal places).

Answer:

\(9.28\)