Base of prism

The base of the perpendicular prism is a rectangular triangle whose legs lengths 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 answer:

V =  540 cm³

Step-by-step explanation:

$$\begin{gathered} S=468{\text{cm}}^2 \\ a:b=3:4 \\ h=b-2 \\ {a}^2+b^2=c^2 \\ a=3x \\ b=4x \\ c=5x \\ {3}^2+4^2=5^2 \\ S=ab+(a+b+c)h \\ S=3\cdot 4\cdot x^2+(3x+4x+5x)(4x-2) \\ 3\cdot 4\cdot x^2+(3x+4x+5x)(4x-2)=468 \\ 3\cdot 4\cdot x^2+(3x+4x+5x)(4x-2)=468 \\ 60x^2-24x-468=0 \\ 60=2^2\cdot 3\cdot 5 \\ 24=2^3\cdot 3 \\ 468=2^2\cdot 3^2\cdot 13 \\ \text{GCD}(60,24,468)=2^2\cdot 3=12 \\ 5x^2-2x-39=0 \\ a=5;b=-2;c=-39 \\ D=b^2-4ac=2^2-4\cdot 5\cdot (-39)=784 \\ D>0 \\ {x}_{1,2}=\frac{-b\pm \sqrt{D}}{2a}=\frac{2\pm \sqrt{784}}{10} \\ {x}_{1,2}=\frac{2\pm 28}{10} \\ {x}_{1,2}=0.2\pm 2.8 \\ {x}_1=3 \\ {x}_2=-2.6 \\ x>0 \\ x=x_1=3\text{cm} \\ a=3\cdot x=3\cdot 3=9\text{cm} \\ b=4\cdot x=4\cdot 3=12\text{cm} \\ c=5\cdot x=5\cdot 3=15\text{cm} \\ h=b-2=12-2=10\text{cm} \\ {S}_2=a\cdot b+(a+b+c)\cdot h=9\cdot 12+(9+12+15)\cdot 10=468{\text{cm}}^2 \\ {S}_2=S \\ V=\frac{a\cdot b}{2}\cdot h=\frac{9\cdot 12}{2}\cdot 10=540{\text{cm}}^3 \end{gathered}$$

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