Question
The volume of a prism is the product of its height and the area of its base, V = B h.
Given: V = 16y^4 + 16y^3 + 48y^2 cubic units.
Which could be the base area and height of the prism?
A. Base area: 4y square units; Height: 4y^2 + 4y + 12 units
B. Base area: 8y^2 square units; Height: y^2 + 2y + 4 units
C. Base area: 12y square units; Height: 4y^2 + 4y + 36 units
D. Base area: 16y^2 square units; Height: y^2 + y + 3 units
Given: V = 16y^4 + 16y^3 + 48y^2 cubic units.
Which could be the base area and height of the prism?
A. Base area: 4y square units; Height: 4y^2 + 4y + 12 units
B. Base area: 8y^2 square units; Height: y^2 + 2y + 4 units
C. Base area: 12y square units; Height: 4y^2 + 4y + 36 units
D. Base area: 16y^2 square units; Height: y^2 + y + 3 units
Answers
GPT-4o mini
To solve this problem, we need to find compatible pairs of base areas and heights that would yield the given volume \( V = 16y^4 + 16y^3 + 48y^2 \).
Using the formula for the volume of a prism, \( V = B \cdot h \), we can express this as \( B \cdot h = 16y^4 + 16y^3 + 48y^2 \).
To find possible pairs, we can use the given base areas and heights and check if their product equals the volume:
**A. Base area: \( B = 4y \) square units; Height: \( h = 4y^2 + 4y + 12 \) units**
\[
V = B \cdot h = 4y \cdot (4y^2 + 4y + 12) = 4y \cdot 4y^2 + 4y \cdot 4y + 4y \cdot 12
\]
Calculating each term:
- \( 4y \cdot 4y^2 = 16y^3 \)
- \( 4y \cdot 4y = 16y^2 \)
- \( 4y \cdot 12 = 48y \)
Thus:
\[
V = 16y^3 + 16y^2 + 48y
\]
This is **not** equal to the volume \( 16y^4 + 16y^3 + 48y^2 \).
---
**B. Base area: \( B = 8y^2 \) square units; Height: \( h = y^2 + 2y + 4 \) units**
\[
V = B \cdot h = 8y^2 \cdot (y^2 + 2y + 4) = 8y^2 \cdot y^2 + 8y^2 \cdot 2y + 8y^2 \cdot 4
\]
Calculating each term:
- \( 8y^2 \cdot y^2 = 8y^4 \)
- \( 8y^2 \cdot 2y = 16y^3 \)
- \( 8y^2 \cdot 4 = 32y^2 \)
Thus:
\[
V = 8y^4 + 16y^3 + 32y^2
\]
This is **not** equal to the volume \( 16y^4 + 16y^3 + 48y^2 \).
---
**C. Base area: \( B = 12y \) square units; Height: \( h = 4y^2 + 4y + 36 \) units**
\[
V = B \cdot h = 12y \cdot (4y^2 + 4y + 36) = 12y \cdot 4y^2 + 12y \cdot 4y + 12y \cdot 36
\]
Calculating each term:
- \( 12y \cdot 4y^2 = 48y^3 \)
- \( 12y \cdot 4y = 48y^2 \)
- \( 12y \cdot 36 = 432y \)
Thus:
\[
V = 48y^3 + 48y^2 + 432y
\]
This is **not** equal to the volume \( 16y^4 + 16y^3 + 48y^2 \).
---
**D. Base area: \( B = 16y^2 \) square units; Height: \( h = y^2 + y + 3 \) units**
\[
V = B \cdot h = 16y^2 \cdot (y^2 + y + 3) = 16y^2 \cdot y^2 + 16y^2 \cdot y + 16y^2 \cdot 3
\]
Calculating each term:
- \( 16y^2 \cdot y^2 = 16y^4 \)
- \( 16y^2 \cdot y = 16y^3 \)
- \( 16y^2 \cdot 3 = 48y^2 \)
Thus:
\[
V = 16y^4 + 16y^3 + 48y^2
\]
This **is equal** to the volume \( 16y^4 + 16y^3 + 48y^2 \).
Therefore, the correct answer is:
**D. Base area: 16y^2 square units; Height: y^2 + y + 3 units.**
Using the formula for the volume of a prism, \( V = B \cdot h \), we can express this as \( B \cdot h = 16y^4 + 16y^3 + 48y^2 \).
To find possible pairs, we can use the given base areas and heights and check if their product equals the volume:
**A. Base area: \( B = 4y \) square units; Height: \( h = 4y^2 + 4y + 12 \) units**
\[
V = B \cdot h = 4y \cdot (4y^2 + 4y + 12) = 4y \cdot 4y^2 + 4y \cdot 4y + 4y \cdot 12
\]
Calculating each term:
- \( 4y \cdot 4y^2 = 16y^3 \)
- \( 4y \cdot 4y = 16y^2 \)
- \( 4y \cdot 12 = 48y \)
Thus:
\[
V = 16y^3 + 16y^2 + 48y
\]
This is **not** equal to the volume \( 16y^4 + 16y^3 + 48y^2 \).
---
**B. Base area: \( B = 8y^2 \) square units; Height: \( h = y^2 + 2y + 4 \) units**
\[
V = B \cdot h = 8y^2 \cdot (y^2 + 2y + 4) = 8y^2 \cdot y^2 + 8y^2 \cdot 2y + 8y^2 \cdot 4
\]
Calculating each term:
- \( 8y^2 \cdot y^2 = 8y^4 \)
- \( 8y^2 \cdot 2y = 16y^3 \)
- \( 8y^2 \cdot 4 = 32y^2 \)
Thus:
\[
V = 8y^4 + 16y^3 + 32y^2
\]
This is **not** equal to the volume \( 16y^4 + 16y^3 + 48y^2 \).
---
**C. Base area: \( B = 12y \) square units; Height: \( h = 4y^2 + 4y + 36 \) units**
\[
V = B \cdot h = 12y \cdot (4y^2 + 4y + 36) = 12y \cdot 4y^2 + 12y \cdot 4y + 12y \cdot 36
\]
Calculating each term:
- \( 12y \cdot 4y^2 = 48y^3 \)
- \( 12y \cdot 4y = 48y^2 \)
- \( 12y \cdot 36 = 432y \)
Thus:
\[
V = 48y^3 + 48y^2 + 432y
\]
This is **not** equal to the volume \( 16y^4 + 16y^3 + 48y^2 \).
---
**D. Base area: \( B = 16y^2 \) square units; Height: \( h = y^2 + y + 3 \) units**
\[
V = B \cdot h = 16y^2 \cdot (y^2 + y + 3) = 16y^2 \cdot y^2 + 16y^2 \cdot y + 16y^2 \cdot 3
\]
Calculating each term:
- \( 16y^2 \cdot y^2 = 16y^4 \)
- \( 16y^2 \cdot y = 16y^3 \)
- \( 16y^2 \cdot 3 = 48y^2 \)
Thus:
\[
V = 16y^4 + 16y^3 + 48y^2
\]
This **is equal** to the volume \( 16y^4 + 16y^3 + 48y^2 \).
Therefore, the correct answer is:
**D. Base area: 16y^2 square units; Height: y^2 + y + 3 units.**