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:

$$\begin{gathered} S=468{\text{cm}}^2 \\ a:b=3:4 \\ h=b-2 \\ {a}^2+b^2=c^2 \\ a=3x \\ b=4x \\ c=5x \\ {3}^2+4^2=5^2 \\ S=ab+(a+b+c)h \\ S=3\cdot 4\cdot x^2+(3x+4x+5x)(4x-2) \\ 3\ast 4\ast x^2+(3x+4x+5x)(4x-2)=468 \\ 3\cdot 4\cdot x^2+(3x+4x+5x)(4x-2)=468 \\ 60x^2-24x-468=0 \\ a=60;b=-24;c=-468 \\ D=b^2-4ac=24^2-4\cdot 60\cdot (-468)=112896 \\ D>0 \\ {x}_{1,2}={\displaystyle \frac{-b\pm \sqrt{D}}{2a}}={\displaystyle \frac{24\pm \sqrt{112896}}{120}} \\ {x}_{1,2}={\displaystyle \frac{24\pm 336}{120}} \\ {x}_{1,2}=0.2\pm 2.8 \\ {x}_1=3 \\ {x}_2=-2.6 \\ \text{ Factored form of the equation: } \\ 60(x-3)(x+2.6)=0 \\ x>0 \\ x=x_1=3=3\text{ cm } \\ a=3\cdot x=3\cdot 3=9\text{ cm } \\ b=4\cdot x=4\cdot 3=12\text{ cm } \\ c=5\cdot x=5\cdot 3=15\text{ cm } \\ h=b-2=12-2=10\text{ cm } \\ {S}_2=a\cdot b+(a+b+c)\cdot h=9\cdot 12+(9+12+15)\cdot 10=468{\text{cm}}^2 \\ {S}_2=S \\ V={\displaystyle \frac{a\cdot b}{2}}\cdot h={\displaystyle \frac{9\cdot 12}{2}}\cdot 10=540{\text{cm}}^3 \end{gathered}$$

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