# Cuboid

Cuboid with edge a=16 cm and body diagonal u=45 cm has volume V=11840 cm3. Calculate the length of the other edges.

Result

b =  20 cm
c =  37 cm

#### Solution:

$V = abc \ \\ 11840 = 16 bc \ \\ bc = 740 \ \\ \ \\ u = \sqrt{ a^2+b^2+c^2 } \ \\ \ \\ 45^2 = 16^2 + b^2 + c^2 \ \\ 1769 = b^2 + c^2 \ \\ 1769 = \dfrac{ 547600}{c^2} + c^2 \ \\ c^4-1769 c^2 + 547600 = 0 \ \\ x = c^2 \ \\ \ \\ x^2 -1769x +547600 =0 \ \\ \ \\ a=1; b=-1769; c=547600 \ \\ D = b^2 - 4ac = 1769^2 - 4\cdot 1 \cdot 547600 = 938961 \ \\ D>0 \ \\ \ \\ x_{1,2} = \dfrac{ -b \pm \sqrt{ D } }{ 2a } = \dfrac{ 1769 \pm \sqrt{ 938961 } }{ 2 } \ \\ x_{1,2} = \dfrac{ 1769 \pm 969 }{ 2 } \ \\ x_{1,2} = 884.5 \pm 484.5 \ \\ x_{1} = 1369 \ \\ x_{2} = 400 \ \\ \ \\ \text{ Factored form of the equation: } \ \\ (x -1369) (x -400) = 0 \ \\ \ \\ b>0; c>0 \ \\ \ \\ b = 20 \ \text { cm }$
$c=37 \ \text { cm }$

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