# Base of prism

The base of the perpendicular prism is a rectangular triangle whose legs length are at a 3: 4 ratio. The height of the prism is 2cm smaller than the larger base leg. Determine the volume of the prism if its surface is 468 cm2.

Correct result:

V =  540 cm3

#### Solution:

$S=468 \ \text{cm}^2 \ \\ a:b=3:4 \ \\ h=b-2 \ \\ \ \\ a^2 + b^2=c^2 \ \\ a=3x \ \\ b=4x \ \\ c=5x \ \\ 3^2+4^2=5^2 \ \\ \ \\ \ \\ S=ab + (a+b+c)h \ \\ S=3 \cdot \ 4 \cdot \ x^2 + (3x+4x+5x)(4x-2) \ \\ \ \\ 3*4*x^2 + (3x+4x+5x)(4x-2)=468 \ \\ \ \\ 3 \cdot \ 4 \cdot \ x^2 + (3x+4x+5x)(4x-2)=468 \ \\ 60x^2 -24x -468=0 \ \\ \ \\ a=60; b=-24; c=-468 \ \\ D=b^2 - 4ac=24^2 - 4\cdot 60 \cdot (-468)=112896 \ \\ D>0 \ \\ \ \\ x_{1,2}=\dfrac{ -b \pm \sqrt{ D } }{ 2a }=\dfrac{ 24 \pm \sqrt{ 112896 } }{ 120 } \ \\ x_{1,2}=\dfrac{ 24 \pm 336 }{ 120 } \ \\ x_{1,2}=0.2 \pm 2.8 \ \\ x_{1}=3 \ \\ x_{2}=-2.6 \ \\ \ \\ \text{ Factored form of the equation: } \ \\ 60 (x -3) (x +2.6)=0 \ \\ \ \\ x>0 \ \\ x=x_{1}=3 \ \text{cm} \ \\ \ \\ a=3 \cdot \ x=3 \cdot \ 3=9 \ \text{cm} \ \\ b=4 \cdot \ x=4 \cdot \ 3=12 \ \text{cm} \ \\ c=5 \cdot \ x=5 \cdot \ 3=15 \ \text{cm} \ \\ h=b-2=12-2=10 \ \text{cm} \ \\ \ \\ S_{2}=a \cdot \ b + (a+b+c) \cdot \ h=9 \cdot \ 12 + (9+12+15) \cdot \ 10=468 \ \text{cm}^2 \ \\ S_{2}=S \ \\ \ \\ V=\dfrac{ a \cdot \ b }{ 2 } \cdot \ h=\dfrac{ 9 \cdot \ 12 }{ 2 } \cdot \ 10=540 \ \text{cm}^3$

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