Question

the volume of a prism is the product of its height and area of its base, $v = bh$. a rectangular prism has a volume of $16y^4 + 16y^3 + 48y^2$ cubic units. which could be the base area and height of the prism?
a base area of $4y$ square units and height of $4y^2 + 4y + 12$ units
a base area of $8y^2$ square units and height of $y^2 + 2y + 4$ units
a base area of $12y$ square units and height of $4y^2 + 4y + 36$ units
a base area of $16y^2$ square units and height of $y^2 + y + 3$ units

Explanation:

Step1: Recall volume formula

$V = Bh$, so we need to check which $B \times h$ equals $16y^4 + 16y^3 + 48y^2$.

Step2: Test Option A

Multiply base area and height:
$4y \times (4y^2 + 4y + 12) = 4y \times 4y^2 + 4y \times 4y + 4y \times 12 = 16y^3 + 16y^2 + 48y$
This does not match the given volume.

Step3: Test Option B

Multiply base area and height:
$8y^2 \times (y^2 + 2y + 4) = 8y^2 \times y^2 + 8y^2 \times 2y + 8y^2 \times 4 = 8y^4 + 16y^3 + 32y^2$
This does not match the given volume.

Step4: Test Option C

Multiply base area and height:
$12y \times (4y^2 + 4y + 36) = 12y \times 4y^2 + 12y \times 4y + 12y \times 36 = 48y^3 + 48y^2 + 432y$
This does not match the given volume.

Step5: Test Option D

Multiply base area and height:
$16y^2 \times (y^2 + y + 3) = 16y^2 \times y^2 + 16y^2 \times y + 16y^2 \times 3 = 16y^4 + 16y^3 + 48y^2$
This matches the given volume.

Answer:

D. a base area of $16y^2$ square units and height of $y^2 + y + 3$ units