The volume of a prism is the product of its height and area of its base, [tex]$V=B h$[/tex]. A rectangular prism has a volume of [tex]$16 y^4+16 y^3+48 y^2$[/tex] cubic units. Which could be the base area and height of the prism?
A. a base area of [tex]$4 y$[/tex] square units and height of [tex]$4 y^2+4 y+12$[/tex] units
B. a base area of [tex]$8 y^2$[/tex] square units and height of [tex]$y^2+2 y+4$[/tex] units
C. a base area of [tex]$12 y$[/tex] square units and height of [tex]$4 y^2+4 y+36$[/tex] units
D. a base area of [tex]$16 y^2$[/tex] square units and height of [tex]$y^2+y+3$[/tex] units
Answer (1)
Multiply the base area and height for each option.
Compare the result with the given volume 16 y 4 + 16 y 3 + 48 y 2 .
Option d: ( 16 y 2 ) ( y 2 + y + 3 ) = 16 y 4 + 16 y 3 + 48 y 2 , which matches the given volume.
The correct option is d: base area of 16 y 2 and height of y 2 + y + 3
Explanation
Understanding the Problem We are given the volume of a rectangular prism as V = 16 y 4 + 16 y 3 + 48 y 2 and asked to find which of the given options for base area B and height h satisfy the formula V = B h . We will test each option by multiplying the given base area and height and comparing the result to the given volume.
Testing Option A Option a: B = 4 y and h = 4 y 3 + 4 y + 12 . Then B h = ( 4 y ) ( 4 y 2 + 4 y + 12 ) = 16 y 3 + 16 y 2 + 48 y . This does not match the given volume V = 16 y 4 + 16 y 3 + 48 y 2 .
Testing Option B Option b: B = 8 y 2 and h = y 2 + 2 y + 4 . Then B h = ( 8 y 2 ) ( y 2 + 2 y + 4 ) = 8 y 4 + 16 y 3 + 32 y 2 . This does not match the given volume V = 16 y 4 + 16 y 3 + 48 y 2 .
Testing Option C Option c: B = 12 y and h = 4 y 2 + 4 y + 36 . Then B h = ( 12 y ) ( 4 y 2 + 4 y + 36 ) = 48 y 3 + 48 y 2 + 432 y . This does not match the given volume V = 16 y 4 + 16 y 3 + 48 y 2 .
Testing Option D Option d: B = 16 y 2 and h = y 2 + y + 3 . Then B h = ( 16 y 2 ) ( y 2 + y + 3 ) = 16 y 4 + 16 y 3 + 48 y 2 . This matches the given volume V = 16 y 4 + 16 y 3 + 48 y 2 .
Final Answer Therefore, the correct option is d, where the base area is 16 y 2 square units and the height is y 2 + y + 3 units.
Examples
Understanding polynomial factorization and volume calculations is crucial in fields like architecture and engineering. For instance, when designing a building, architects need to calculate the volume of different sections to estimate material requirements. If the volume is expressed as a polynomial, factoring it can help determine possible dimensions of the structure, ensuring efficient use of space and resources. This also applies to designing containers or storage units where optimizing volume based on polynomial expressions is essential for cost-effectiveness.
