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The volume of a prism is the product of its height and the area of its base, [tex]V = B h[/tex].

A rectangular prism has a volume of [tex]16y^4 + 16y^3 + 48y^2[/tex] cubic units. Which could be the base area and height of the prism?

A. A base area of [tex]4y[/tex] square units and a height of [tex]4y^2 + 4y + 12[/tex] units
B. A base area of [tex]8y^2[/tex] square units and a height of [tex]y^2 + 2y + 4[/tex] units
C. A base area of [tex]12y[/tex] square units and a height of [tex]4y^2 + 4y + 36[/tex] units
D. A base area of [tex]16y^2[/tex] square units and a height of [tex]y^2 + y + 3[/tex] units

Sagot :

GinnyAnsweridn
To identify the correct base area ([tex]\(B\)[/tex]) and height ([tex]\(h\)[/tex]) for the given prism volume [tex]\(16y^4 + 16y^3 + 48y^2\)[/tex], we use the formula for the volume of a prism:
[tex]\[ V = B \cdot h \][/tex]

Given the volume:
[tex]\[ V = 16y^4 + 16y^3 + 48y^2 \][/tex]

We need to check each pair of base area and height combinations to see if their product equals the given volume.

### Examining Each Choice

1. Base Area: [tex]\(4y\)[/tex] [tex]\( \text{square units} \)[/tex]
Height: [tex]\(4y^2 + 4y + 12\)[/tex] [tex]\( \text{units} \)[/tex]

Calculate:
[tex]\[ B \cdot h = 4y (4y^2 + 4y + 12) = 4y \cdot 4y^2 + 4y \cdot 4y + 4y \cdot 12 = 16y^3 + 16y^2 + 48y \][/tex]

This result [tex]\(16y^3 + 16y^2 + 48y\)[/tex] is not equal to the given volume [tex]\(16y^4 + 16y^3 + 48y^2\)[/tex].

2. Base Area: [tex]\(8y^2\)[/tex] [tex]\( \text{square units} \)[/tex]
Height: [tex]\(y^2 + 2y + 4\)[/tex] [tex]\( \text{units} \)[/tex]

Calculate:
[tex]\[ B \cdot h = 8y^2 (y^2 + 2y + 4) = 8y^2 \cdot y^2 + 8y^2 \cdot 2y + 8y^2 \cdot 4 = 8y^4 + 16y^3 + 32y^2 \][/tex]

The result [tex]\(8y^4 + 16y^3 + 32y^2\)[/tex] is not equal to the given volume [tex]\(16y^4 + 16y^3 + 48y^2\)[/tex].

3. Base Area: [tex]\(12y\)[/tex] [tex]\( \text{square units} \)[/tex]
Height: [tex]\(4y^2 + 4y + 36\)[/tex] [tex]\( \text{units} \)[/tex]

Calculate:
[tex]\[ B \cdot h = 12y (4y^2 + 4y + 36) = 12y \cdot 4y^2 + 12y \cdot 4y + 12y \cdot 36 = 48y^3 + 48y^2 + 432y \][/tex]

The result [tex]\(48y^3 + 48y^2 + 432y\)[/tex] is not equal to the given volume [tex]\(16y^4 + 16y^3 + 48y^2\)[/tex].

4. Base Area: [tex]\(16y^2\)[/tex] [tex]\( \text{square units} \)[/tex]
Height: [tex]\(y^2 + y + 3\)[/tex] [tex]\( \text{units} \)[/tex]

Calculate:
[tex]\[ B \cdot h = 16y^2 (y^2 + y + 3) = 16y^2 \cdot y^2 + 16y^2 \cdot y + 16y^2 \cdot 3 = 16y^4 + 16y^3 + 48y^2 \][/tex]

The result [tex]\(16y^4 + 16y^3 + 48y^2\)[/tex] equals the given volume [tex]\(16y^4 + 16y^3 + 48y^2\)[/tex].

### Conclusion

The correct pair of base area and height that results in the volume [tex]\(16y^4 + 16y^3 + 48y^2\)[/tex] is:
- Base Area: [tex]\(16y^2\)[/tex] square units
- Height: [tex]\(y^2 + y + 3\)[/tex] units
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