A pyramid has a rectangular base of length (3x + 1)cm and width xcm.
It also has a perpendicular height of 12 cm.
The volume of the pyramid is 96 cmº.
Given that the volume of a pyramid is one third of the area of the base
multiplied by the perpendicular height, find the dimensions of the base
of the pyramid.

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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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

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step-by-step explanation:

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