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The volume of a rectangular prism (shown below) is 48x^3+56x^2+16x 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 volume of 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. Show all work.

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MrRoyal

Answer:

(a)

[tex]Length = 8x\\Width = 3x + 2\\Height = 2x + 1[/tex]

(b)

[tex]P(2) = 640[/tex]

(c)

[tex]Length= 16[/tex]

[tex]Width = 8[/tex]

[tex]Height =5[/tex]

(d)

[tex]Volume = 640[/tex]

Step-by-step explanation:

Given

[tex]P(x) = 48x^3 + 56x^2 + 16x[/tex]

Solving (a): The prism dimension

We have:

[tex]P(x) = 48x^3 + 56x^2 + 16x[/tex]

Factor out 8x

[tex]P(x) = 8x(6x^2 + 7x + 2)[/tex]

Expand 7x

[tex]P(x) = 8x(6x^2 + 4x + 3x + 2)[/tex]

Factorize

[tex]P(x) = 8x(2x(3x + 2) +1( 3x + 2))[/tex]

Factor out 3x + 2

[tex]P(x) = 8x(3x + 2)(2x + 1)[/tex]

So, the dimensions are:

[tex]Length = 8x\\Width = 3x + 2\\Height = 2x + 1[/tex]

Solving (b): The volume when [tex]x = 2[/tex]

We have:

[tex]P(x) = 48x^3 + 56x^2 + 16x[/tex]

[tex]P(2) = 48 * 2^3 + 56 * 2^2 + 16 * 2[/tex]

[tex]P(2) = 640[/tex]

Solving (c): The dimensions when [tex]x = 2[/tex]

We have:

[tex]Length = 8x\\Width = 3x + 2\\Height = 2x + 1[/tex]

Substitute 2 for x

[tex]Length=8*2[/tex]

[tex]Length= 16[/tex]

[tex]Width = 3*2+2[/tex]

[tex]Width = 8[/tex]

[tex]Height = 2*2 + 1[/tex]

[tex]Height =5[/tex]

So, we have:

[tex]Length= 16[/tex]

[tex]Width = 8[/tex]

[tex]Height =5[/tex]

Solving (d), the volume in (c)

We have:

[tex]Volume = Length * Width * Height[/tex]

[tex]Volume = 16 * 8 * 5[/tex]

[tex]Volume = 640[/tex]

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