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:

S = 468 cm^{2} a : b = 3 : 4 h = b - 2 a^{2} + b^{2} = c^{2} a = 3 x b = 4 x c = 5 x 3^{2} + 4^{2} = 5^{2} S = a b + (a + b + c) h S = 3 \cdot 4 \cdot x^{2} + (3 x + 4 x + 5 x) (4 x - 2) 3 \cdot 4 \cdot x^{2} + (3 x + 4 x + 5 x) (4 x - 2) = 468 3 \cdot 4 \cdot x^{2} + (3 x + 4 x + 5 x) (4 x - 2) = 468 60 x^{2} - 24 x - 468 = 0 60 = 2^{2} \cdot 3 \cdot 5 24 = 2^{3} \cdot 3 468 = 2^{2} \cdot 3^{2} \cdot 13 GCD (60, 24, 468) = 2^{2} \cdot 3 = 12 5 x^{2} - 2 x - 39 = 0 a = 5; b = - 2; c = - 39 D = b^{2} - 4 a c = 2^{2} - 4 \cdot 5 \cdot (- 39) = 784 D > 0 x_{1, 2} = \frac{- b \pm D}{2 a} = \frac{2 \pm 7 8 4}{1 0} x_{1, 2} = \frac{2 \pm 2 8}{1 0} x_{1, 2} = 0.2 \pm 2.8 x_{1} = 3 x_{2} = - 2.6 x > 0 x = x_{1} = 3 cm a = 3 \cdot x = 3 \cdot 3 = 9 cm b = 4 \cdot x = 4 \cdot 3 = 12 cm c = 5 \cdot x = 5 \cdot 3 = 15 cm h = b - 2 = 12 - 2 = 10 cm S_{2} = a \cdot b + (a + b + c) \cdot h = 9 \cdot 12 + (9 + 12 + 15) \cdot 10 = 468 cm^{2} S_{2} = S V = \frac{a \cdot b}{2} \cdot h = \frac{9 \cdot 1 2}{2} \cdot 10 = 540 cm^{3}

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