Answer:
The dimension of the base of the Rectangle Pyramid is Length = 10 and Width = 8/3
Step-by-step explanation:
Given
Rectangle Pyramid
Base Length = 3x + 1
Base Width = x
Height = 12
Volume = 96
Required
Dimension of the base of the pyramid
Given that the volume of the pyramid is ⅓ of the base area * the height.
This is represented mathematical as
Volume = ⅓ * base area * height.
Where
Base area = width * length
Base area = (3x + 1) * x
Base area = 3x² + x.
So,
Volume becomes
Volume = ⅓ * (3x² + x) * 12.
Volume = (3x² + x) * 4
Substitute 96 for volume
96 = (3x² + x) * 4
Divide both sides by 4
96/4 = (3x² + x) * 4/4
24 = 3x² + x
Subtract 24 fr both sides
24 - 24 = 3x² + x - 24
0 = 3x² + x - 24
3x² + x - 24 = 0
Expand
3x² + 9x - 8x - 24 = 0
Factorize
3x(x + 3) - 8(x + 3) = 0
(3x - 8)(x + 3) = 0
3x - 8 = 0 or x + 3 = 0
3x = 8 or x = -3
x = 8/3 or x = -3
Recall that
Length = 3x + 1
Width = x
For any of the above expression, x can't be less than 0; so, x = -3 can't be considered.
Substitute x = 8/3
Length = 3x + 1
Length = 3(8/3) + 1
Length = 8 + 1
Length = 9
Width = x
Width = 8/3
Hence, the dimension of the base of the Rectangle Pyramid is Length = 10 and Width = 8/3
Based on the information given the dimensions of the base of the pyramid are: Width 8/3cm or 2 ²/3cm ; Length 9cm.
Given:
Length=(3x + 1)cm
Width=x cm
Perpendicular height=12cm
Volume of the pyramid=96 cmº
Hence:
Area of the base=(3x + 1)x=(3x² +x) (cm²)
Volume of the pyramid=1/3×(3x² +x)×12=96
96=(3x+1)(x)(12)/3
96=(3x+1)(x)(4)
96/4=(3x+1)(x)
24=3x²+x
3x²+x-24=0
(3x-8)(x+3)=0
3x-8=0
3x=8
x=8/3 or 2 ²/3cm width
Length=3(8/3)+1
Length=8+1
Length=9cm
Inconclusion the dimensions of the base of the pyramid are: Width 8/3cm or 2 ²/3cm ; Length 9cm.
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