# Cuboid

Cuboid with edge a=6 cm and body diagonal u=31 cm has volume V=900 cm3. Calculate the length of the other edges.

Correct result:

b =  30 cm
c =  5 cm

#### Solution:

$a=6 \ \text{cm} \ \\ V=900 \ \text{cm}^3 \ \\ u=31 \ \text{cm} \ \\ \ \\ V=abc \ \\ 900=6 \ bc \ \\ bc=150 \ \\ \ \\ u=\sqrt{ a^2+b^2+c^2 } \ \\ \ \\ 31^2=6^2 + b^2 + c^2 \ \\ \ \\ 925=b^2 + c^2 \ \\ 925=\dfrac{ 22500 }{ c^2 } + c^2 \ \\ c^4-925 \ c^2 + 22500=0 \ \\ x=c^2 \ \\ \ \\ x^2-925x + 22500=0 \ \\ \ \\ x^2 -925x +22500=0 \ \\ \ \\ a=1; b=-925; c=22500 \ \\ D=b^2 - 4ac=925^2 - 4\cdot 1 \cdot 22500=765625 \ \\ D>0 \ \\ \ \\ x_{1,2}=\dfrac{ -b \pm \sqrt{ D } }{ 2a }=\dfrac{ 925 \pm \sqrt{ 765625 } }{ 2 } \ \\ x_{1,2}=\dfrac{ 925 \pm 875 }{ 2 } \ \\ x_{1,2}=462.5 \pm 437.5 \ \\ x_{1}=900 \ \\ x_{2}=25 \ \\ \ \\ \text{ Factored form of the equation: } \ \\ (x -900) (x -25)=0 \ \\ \ \\ c_{1}=\sqrt{ x_{1} }=\sqrt{ 900 }=30 \ \\ c_{2}=\sqrt{ x_{2} }=\sqrt{ 25 }=5 \ \\ \ \\ b=c_{1}=30 \ \text{cm}$

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