Answer:
a) Length= 8x
Width = 3x+2
Height = 2x+1
b) The volume of Prism = 640
c) Dimensions of prism are:
Length= 16
Width = 8
Height = 5
d) Volume of Prism is 640
Step-by-step explanation:
Volume of Prism = [tex]48x^3+56x^2+16x[/tex]
a) What are the dimensions of prism.
Dimensions of prism can be found by finding the factors of the given Volume.
Volume of Prism = [tex]48x^3+56x^2+16x[/tex]
Take 8x common
[tex]8x(6x^2+7x+2)[/tex]
Now we will factor the term: [tex]6x^2+7x+2[/tex]
[tex]6x^2+3x+4x+2\\=3x(2x+1)+2(2x+1)\\=(3x+2)(2x+1)[/tex]
So, factors of term : [tex]6x^2+7x+2[/tex] are [tex](3x+2)(2x+1)[/tex]
Replace term with their factors
[tex]8x(6x^2+7x+2)\\=8x(3x+2)(2x+1)[/tex]
We know that [tex]Volume\: of\: Prism = Length \times Width \times Height[/tex]
So, [tex]48x^3+56x^2+16x=8x(3x+2)(2x+1)[/tex]
So, Length= 8x
Width = 3x+2
Height = 2x+1
b) if x =2, use the polynomial [tex]48x^3+56x^2+16x[/tex] to find Volume of prism.
Put x =2 and calculate Volume
Volume of Prism = [tex]48x^3+56x^2+16x[/tex]
Volume of Prism = [tex]48(2)^3+56(2)^2+16(2)\\=48(8)+56(4)+32\\=384+224+32\\=640[/tex]
So, The volume of Prism = 640
c) If x =2, Use the factors found in part a) to calculate the dimensions.
The dimensions are:
Length= 8x
Width = 3x+2
Height = 2x+1
Put x =2 and find answers
Length= 8x = 8(2) = 16
Width = 3x+2 = 3(2)+2 = 6+2 =8
Height = 2x+1 = 2(2)+1 = 4+1 = 5
So, Dimensions of prism are:
Length= 16
Width = 8
Height = 5
d) Using the dimensions found in part c, calculate the volume.
We know that [tex]Volume\: of\: Prism = Length \times Width \times Height[/tex]
And
Length= 16
Width = 8
Height = 5
Finding Volume:
[tex]Volume\: of\: Prism = Length \times Width \times Height[/tex]
[tex]Volume\:of\:Prism=16\times 8 \times 5\\Volume\:of\:Prism=640[/tex]
So, Volume of Prism is 640
