Question

how are they related? here are two formulas for the volume of a cylinder: $v = \pi r^2 \cdot h$ $v = b \cdot h$ describe how the formulas are related. use the sketch tool if it helps you to show your thinking.

Explanation:

The formula \(V = \pi r^{2}\cdot h\) is specific to cylinders. Here, \(\pi r^{2}\) represents the area of the circular base of the cylinder. In the general formula for the volume of a prism - like solid \(V=B\cdot h\) (where \(B\) is the area of the base and \(h\) is the height), for a cylinder, the base \(B\) is a circle. So, when \(B = \pi r^{2}\) (the area of the circular base), substituting \(B\) into \(V = B\cdot h\) gives the formula \(V=\pi r^{2}\cdot h\).

Answer:

The two formulas are related because in the general volume formula \(V = B\cdot h\) (for prisms and cylinder - like solids), when the base \(B\) is a circle (for a cylinder), \(B=\pi r^{2}\). Substituting \(B = \pi r^{2}\) into \(V = B\cdot h\) results in the formula \(V=\pi r^{2}\cdot h\) which is the volume formula for a cylinder.