Question

The volume of a rectangular prism is 48x^3+56x^2+16 X. Answer the following questions 1. What are the dimensions of the prism?2. If x=2, use the polynomial 48x^3+56x^2+16x to find the prism. 3. If x=2, use the factors found in part a to calculate each dimension. 4. Using the dimensions found in part c, calculate the volume.

Answer 1

(1) From the statement, we know that:

• the volume of a rectangular prism is:

`V(x)=48x^3+56x^2+16x\text{.}`

• and it also can be computed by:

`V=a\cdot b\cdot c.`

Where a, b and c are the lengths of the sides.

By factoring in the expression of V(x), we get:

`\begin{gathered} V(x)=(48x^3+56x^2+16x) \\ =(8x)\cdot(6x^2+7x+2) \\ =(8x)\cdot(3x+2)\cdot(2x+1)\text{.} \end{gathered}`

Comparing this result with the expression above, we see that the sides of the prism are:

`\begin{gathered} a=8x, \\ b=3x+2, \\ c=2x+1. \end{gathered}`

(2) Relacing the value x = 2 in the polynomial V(x), we get:

`V(2)=48\cdot2^3+56\cdot2^2+16\cdot2=640.`

(3) Replacing the value x = 2 in the expressions for the dimensions found in point (2), we get:

`\begin{gathered} a=8\cdot2=16, \\ b=3\cdot2+2=8, \\ c=2\cdot2+1=5. \end{gathered}`

(4) Replacing the dimensions from point (3) in the expression for the volume, we get:

`V=16\cdot8\cdot5=640.`

Which is the same result that we get in point (2), as it should be.

Answer

(1) Dimensions

`\begin{gathered} a=8x, \\ b=3x+2, \\ c=2x+1. \end{gathered}`

(2) Volume when x = 2:

`V(2)=640.`

(3) Dimensions when x = 2:

`\begin{gathered} a=16, \\ b=8, \\ c=5. \end{gathered}`

(4) Volume when x = 2:

`V=16\cdot8\cdot5=640.`