Question

for each pair of solids, determine if their volumes are the same or different. if the volumes are different, identify the solid with the greatest volume. explain your reasoning. question 9 part a a prism and a pyramid have the same height. the pyramids base has 3 times the area of the prisms base. edit view insert format tools table 12pt paragraph b i u a - 2 1 1 their volumes are the same question 9 part b a pyramid and a cylinder have bases with the same area. the cylinders height is 3 times that of the pyramid. edit view insert format tools table 12pt paragraph b i u a - 2 1 1 the cylinder has a greater volume. question 9 part c a cone and a cylinder have the same height. the cones radius is 3 times the length of the cylinders radius. edit view insert format tools table 12pt paragraph b i u a - 2 1 1

Explanation:

Step1: Recall volume formulas

The volume formula for a prism is \(V_{prism}=B_{prism}h\), for a pyramid \(V_{pyramid}=\frac{1}{3}B_{pyramid}h\), for a cylinder \(V_{cylinder}=B_{cylinder}h=\pi r_{cylinder}^2h\), and for a cone \(V_{cone}=\frac{1}{3}\pi r_{cone}^2h\).

Step2: Analyze Part A

Let \(B_{prism}=B\), then \(B_{pyramid} = 3B\) and \(h_{prism}=h_{pyramid}=h\).
\(V_{prism}=Bh\), \(V_{pyramid}=\frac{1}{3}(3B)h=Bh\).

Step3: Analyze Part B

Let \(B_{pyramid}=B_{cylinder}=B\), \(h_{pyramid}=h\), then \(h_{cylinder}=3h\).
\(V_{pyramid}=\frac{1}{3}Bh\), \(V_{cylinder}=B(3h) = 3Bh\).

Step4: Analyze Part C

Let \(h_{cone}=h_{cylinder}=h\), \(r_{cylinder}=r\), then \(r_{cone}=3r\).
\(V_{cone}=\frac{1}{3}\pi(3r)^2h = 3\pi r^2h\), \(V_{cylinder}=\pi r^2h\).

Answer:

• Part A: Their volumes are the same.
• Part B: The cylinder has a greater volume.
• Part C: The cone has a greater volume.