Question

Given a rectangular pyramid and a rectangular prism that have the same base and same height, how do their volumes compare? If the pyramid was full of water, how much of the prism would it fill up? Name another pair of three-dimensional objects that have a relationship similar to this.

Answer 1

Answer:

Part 1) The volume of the prism is 3 times greater than the volume of the pyramid

Part 2) The prism would fill only one-third

Part 3) The cylinder and the cone

Step-by-step explanation:

Part 1) How do their volumes compare?

we know that

The volume of a rectangular prism is equal to

$Vprism=BH$

where

B is the area of the base of rectangular prism

H is the height of the prism

The volume of a rectangular pyramid is equal to

$Vpyramid=\frac{1}{3}BH$

where

B is the area of the base of rectangular pyramid

H is the height of the pyramid

Compare the volumes

$Vprism=BH$ -----> equation A

$Vpyramid=\frac{1}{3}BH$ -----> equation B

substitute equation A in equation B

$Vpyramid=\frac{1}{3}Vprism$

$Vprism=(3)Vpyramid$

The volume of the prism is 3 times greater than the volume of the pyramid

Part 2) If the pyramid was full of water, how much of the prism would it fill up?

Remember that

$Vpyramid=\frac{1}{3}Vprism$

therefore

The prism would fill only one-third

Part 3) Name another pair of three-dimensional objects that have a relationship similar to this

we have the cylinder and the cone

The volume of the cylinder is equal to

$Vcylinder=\pi r^{2}h$

The volume of the cone is equal to

$Vcone=\frac{1}{3}\pi r^{2}h$

Compare the volumes

$Vcylinder=(3)Vcone$

The volume of the cylinder is 3 times greater than the volume of the cone