Question

find the formula for the volume of the region outside the smaller rectangular solid and inside the larger rectangular solid. then express the volume in factored form.
the unfactored formula for the volume of the shaded region is . (do not factor.)

Explanation:

Step1: Find volume of larger solid

The volume of a rectangular - solid is given by \(V = l\times w\times h\). For the larger rectangular solid with length \(a\), width \(a\), and height \(a\), its volume \(V_{1}=a\times a\times a=a^{3}\).

Step2: Find volume of smaller solid

For the smaller rectangular solid with length \(b\), width \(b\), and height \(b\), its volume \(V_{2}=b\times b\times b = b^{3}\).

Step3: Find volume of the region

The volume of the region outside the smaller solid and inside the larger solid is the difference between the volume of the larger solid and the volume of the smaller solid. So \(V = V_{1}-V_{2}=a^{3}-b^{3}\).

Answer:

\(a^{3}-b^{3}\)