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 16y4 + 16y3 + 48y2 cubic units. Which could be the base area and height of the prism?

a base area of 4y square units and height of 4y2 + 4y + 12 units
a base area of 8y2 square units and height of y2 + 2y + 4 units
a base area of 12y square units and height of 4y2 + 4y + 36 units
a base area of 16y2 square units and height of y2 + y + 3 units

Answer 1

Option 4) a base area of
`16y^2` square units and height of
`y^2 + y + 3` units

Explanation:

We are given the following in the question:

Volume of prism = Bh

where B is the area of the base and h is heigth of the prism

Volume of prism =

`16y^4 + 16y^3 + 48y^2 \text{ cubic units}`

We have to find the base area and the height of the prism from the given options.

1)

Base area =
`4y`

Height =
`4y^2 + 4y + 12`

Volume =
`4y(4y^2 + 4y + 12) = 16y^3 + 16y^2 +48y`

which is not equal to the given volume

2)

Base area =
`8y^2`

Height =
`y^2 + 2y + 4`

Volume =
`8y^2(y^2 + 2y + 4) = 8y^4 + 16y^3 +32y^2`

which is not equal to the given volume

3)

Base area =
`12y`

Height =
`4y^2 + 4y + 36`

Volume =
`12y(4y^2 + 4y + 36) = 48y^3 + 48y^2 +432y`

which is not equal to the given volume

4)

Base area =
`16y^2`

Height =
`y^2 + y + 3`

Volume =
`16y^2(y^2 + y + 3) = 16y^4 + 16y^3 +48y^2`

which is equal to the given volume

Answer 2

We have a prism with a volume of 16y⁴ + 16y³ + 48y² cubic units.
Its volume is equal to the area of its base times its height.
Of course, for those to be the base area and height of this prism, they would have to multiply to 16y⁴ + 16y³ + 48y² cubic units.
Let's test each of these answers to see which gives us the correct volume.
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a base area of 4y square units and height of 4y² + 4y + 12 units
We find the volume by multiplying the base area by the height...
4y(4y² + 4y + 12)
Distribute the 4y to each term inside the parentheses.
16y³ + 16y² + 48y
This is not the right volume, so these can not be dimensions of our prism.
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a base area of 8y² square units and height of y² + 2y + 4 units
We find the volume by multiplying the base area by the height...
8y²(y² + 2y + 4)
Distribute the 8y² to each term inside the parentheses.
8y⁴ + 16y³ + 32y²
This is not the right volume, so these can not be dimensions of our prism.
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a base area of 12y square units and height of 4y² + 4y + 36 units
We find the volume by multiplying the base area by the height...
12y(4y² + 4y + 36)
Distribute the 12y to each term inside the parentheses.
48y³ + 48y² + 432y
This is not the right volume, so these can not be dimensions of our prism.
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a base area of 16y² square units and height of y² + y + 3 units
We find the volume by multiplying the base area by the height...
16y²(y² + y + 3)
Distribute the 16y² to each term inside the parentheses.
16y⁴ + 16y³ + 48y²
The volume fits, so these could be the base area and height of our prism.
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D. a base area of 16y² square units and height of y² + y + 3 units
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