Answer :
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.
