Question

1. what is the volume of the smallest square - based prism that would hold this cylinder? 6. a cube has a side length of 8 cm. inside is a hole, also in the shape of a cube, with a side length of 4 cm. how much volume does the part that is still there have?

Explanation:

Question 5:

Step1: Determine the dimensions of the square - based prism

The cylinder has a radius of \(r = 20\space cm\), so the diameter of the cylinder is \(d=2r = 2\times20=40\space cm\). The height of the cylinder is \(h = 50\space cm\). For the smallest square - based prism that can hold the cylinder, the base of the prism is a square with side length equal to the diameter of the cylinder (\(40\space cm\)) and the height of the prism is equal to the height of the cylinder (\(50\space cm\)).

Step2: Calculate the volume of the square - based prism

The volume formula for a square - based prism (a rectangular prism with a square base) is \(V=s^{2}h\), where \(s\) is the side length of the square base and \(h\) is the height. Here, \(s = 40\space cm\) and \(h = 50\space cm\).
So \(V=(40)^{2}\times50=1600\times50 = 80000\space cm^{3}\)

Step1: Calculate the volume of the large cube

The volume formula for a cube is \(V = s^{3}\), where \(s\) is the side length of the cube. For the large cube with side length \(s_{1}=8\space cm\), its volume \(V_{1}=s_{1}^{3}=8^{3}=8\times8\times8 = 512\space cm^{3}\)

Step2: Calculate the volume of the small cube (the hole)

For the small cube with side length \(s_{2} = 4\space cm\), its volume \(V_{2}=s_{2}^{3}=4^{3}=4\times4\times4=64\space cm^{3}\)

Step3: Calculate the remaining volume

The remaining volume \(V = V_{1}-V_{2}\). Substitute \(V_{1}=512\space cm^{3}\) and \(V_{2}=64\space cm^{3}\) into the formula, we get \(V = 512 - 64=448\space cm^{3}\)

Answer:

\(80000\space cm^{3}\)

Question 6: