Welcome to Westonci.ca, where you can find answers to all your questions from a community of experienced professionals. Experience the ease of finding quick and accurate answers to your questions from professionals on our platform. Get immediate and reliable solutions to your questions from a community of experienced professionals on our platform.
Sagot :
To determine which of the given pairs of base area and height correctly represent the volume of the rectangular prism, we'll evaluate each of the provided options step-by-step. We know the volume [tex]\( V \)[/tex] of the prism is given as [tex]\( 16y^4 + 16y^3 + 48y^2 \)[/tex] cubic units. The volume formula for a prism is [tex]\( V = B \cdot h \)[/tex], where [tex]\( B \)[/tex] is the base area and [tex]\( h \)[/tex] is the height.
### Option 1:
Base Area: [tex]\( 4y \)[/tex] square units
Height: [tex]\( 4y^2 + 4y + 12 \)[/tex] units
Calculate the volume:
[tex]\[ V_1 = (4y) \cdot (4y^2 + 4y + 12) \][/tex]
[tex]\[ V_1 = 4y \cdot 4y^2 + 4y \cdot 4y + 4y \cdot 12 \][/tex]
[tex]\[ V_1 = 16y^3 + 16y^2 + 48y \][/tex]
This volume is:
[tex]\[ 16y^3 + 16y^2 + 48y \][/tex]
which does not match the given volume [tex]\( 16y^4 + 16y^3 + 48y^2 \)[/tex].
### Option 2:
Base Area: [tex]\( 8y^2 \)[/tex] square units
Height: [tex]\( y^2 + 2y + 4 \)[/tex] units
Calculate the volume:
[tex]\[ V_2 = (8y^2) \cdot (y^2 + 2y + 4) \][/tex]
[tex]\[ V_2 = 8y^2 \cdot y^2 + 8y^2 \cdot 2y + 8y^2 \cdot 4 \][/tex]
[tex]\[ V_2 = 8y^4 + 16y^3 + 32y^2 \][/tex]
This volume is:
[tex]\[ 8y^4 + 16y^3 + 32y^2 \][/tex]
which does not match the given volume [tex]\( 16y^4 + 16y^3 + 48y^2 \)[/tex].
### Option 3:
Base Area: [tex]\( 12y \)[/tex] square units
Height: [tex]\( 4y^2 + 4y + 36 \)[/tex] units
Calculate the volume:
[tex]\[ V_3 = (12y) \cdot (4y^2 + 4y + 36) \][/tex]
[tex]\[ V_3 = 12y \cdot 4y^2 + 12y \cdot 4y + 12y \cdot 36 \][/tex]
[tex]\[ V_3 = 48y^3 + 48y^2 + 432y \][/tex]
This volume is:
[tex]\[ 48y^3 + 48y^2 + 432y \][/tex]
which does not match the given volume [tex]\( 16y^4 + 16y^3 + 48y^2 \)[/tex].
### Option 4:
Base Area: [tex]\( 16y^2 \)[/tex] square units
Height: [tex]\( y^2 + y + 3 \)[/tex] units
Calculate the volume:
[tex]\[ V_4 = (16y^2) \cdot (y^2 + y + 3) \][/tex]
[tex]\[ V_4 = 16y^2 \cdot y^2 + 16y^2 \cdot y + 16y^2 \cdot 3 \][/tex]
[tex]\[ V_4 = 16y^4 + 16y^3 + 48y^2 \][/tex]
This volume is:
[tex]\[ 16y^4 + 16y^3 + 48y^2 \][/tex]
which does match the given volume [tex]\( 16y^4 + 16y^3 + 48y^2 \)[/tex].
So, the correct answer is:
a base area of [tex]\( 16y^2 \)[/tex] square units and height of [tex]\( y^2 + y + 3 \)[/tex] units.
### Option 1:
Base Area: [tex]\( 4y \)[/tex] square units
Height: [tex]\( 4y^2 + 4y + 12 \)[/tex] units
Calculate the volume:
[tex]\[ V_1 = (4y) \cdot (4y^2 + 4y + 12) \][/tex]
[tex]\[ V_1 = 4y \cdot 4y^2 + 4y \cdot 4y + 4y \cdot 12 \][/tex]
[tex]\[ V_1 = 16y^3 + 16y^2 + 48y \][/tex]
This volume is:
[tex]\[ 16y^3 + 16y^2 + 48y \][/tex]
which does not match the given volume [tex]\( 16y^4 + 16y^3 + 48y^2 \)[/tex].
### Option 2:
Base Area: [tex]\( 8y^2 \)[/tex] square units
Height: [tex]\( y^2 + 2y + 4 \)[/tex] units
Calculate the volume:
[tex]\[ V_2 = (8y^2) \cdot (y^2 + 2y + 4) \][/tex]
[tex]\[ V_2 = 8y^2 \cdot y^2 + 8y^2 \cdot 2y + 8y^2 \cdot 4 \][/tex]
[tex]\[ V_2 = 8y^4 + 16y^3 + 32y^2 \][/tex]
This volume is:
[tex]\[ 8y^4 + 16y^3 + 32y^2 \][/tex]
which does not match the given volume [tex]\( 16y^4 + 16y^3 + 48y^2 \)[/tex].
### Option 3:
Base Area: [tex]\( 12y \)[/tex] square units
Height: [tex]\( 4y^2 + 4y + 36 \)[/tex] units
Calculate the volume:
[tex]\[ V_3 = (12y) \cdot (4y^2 + 4y + 36) \][/tex]
[tex]\[ V_3 = 12y \cdot 4y^2 + 12y \cdot 4y + 12y \cdot 36 \][/tex]
[tex]\[ V_3 = 48y^3 + 48y^2 + 432y \][/tex]
This volume is:
[tex]\[ 48y^3 + 48y^2 + 432y \][/tex]
which does not match the given volume [tex]\( 16y^4 + 16y^3 + 48y^2 \)[/tex].
### Option 4:
Base Area: [tex]\( 16y^2 \)[/tex] square units
Height: [tex]\( y^2 + y + 3 \)[/tex] units
Calculate the volume:
[tex]\[ V_4 = (16y^2) \cdot (y^2 + y + 3) \][/tex]
[tex]\[ V_4 = 16y^2 \cdot y^2 + 16y^2 \cdot y + 16y^2 \cdot 3 \][/tex]
[tex]\[ V_4 = 16y^4 + 16y^3 + 48y^2 \][/tex]
This volume is:
[tex]\[ 16y^4 + 16y^3 + 48y^2 \][/tex]
which does match the given volume [tex]\( 16y^4 + 16y^3 + 48y^2 \)[/tex].
So, the correct answer is:
a base area of [tex]\( 16y^2 \)[/tex] square units and height of [tex]\( y^2 + y + 3 \)[/tex] units.
Thanks for stopping by. We strive to provide the best answers for all your questions. See you again soon. Thank you for your visit. We're committed to providing you with the best information available. Return anytime for more. Get the answers you need at Westonci.ca. Stay informed by returning for our latest expert advice.