Question

for each par 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
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

Explanation:

Question 9 Part A

Step1: Write volume formulas

Volume of prism \(V_p = B_p\times h\) (where \(B_p\) is base area of prism, \(h\) is height). Volume of pyramid \(V_y=\frac{1}{3}B_y\times h\) (where \(B_y\) is base area of pyramid, \(h\) is height). Given \(B_y = 3B_p\).

Step2: Substitute \(B_y\) in pyramid volume formula

\(V_y=\frac{1}{3}(3B_p)\times h=B_p\times h\)

Step3: Compare volumes

Since \(V_p = B_p\times h\) and \(V_y=B_p\times h\), volumes are the same.

Answer:

Their volumes are the same.

Question 9 Part B

Step1: Write volume formulas

Volume of pyramid \(V_y=\frac{1}{3}B\times h_y\) (where \(B\) is base area, \(h_y\) is height of pyramid). Volume of cylinder \(V_c = B\times h_c\) (where \(h_c\) is height of cylinder). Given \(h_c = 3h_y\).

Step2: Substitute \(h_c\) in cylinder volume formula

\(V_c=B\times(3h_y)\)

Step3: Compare volumes

\(V_y=\frac{1}{3}B\times h_y\) and \(V_c = 3B\times h_y\). So \(V_c>V_y\)