# Cuboid - volume and areas

The cuboid has a volume of 250 cm3, a surface of 250 cm2 and one side 5 cm long. How do I calculate the remaining sides?

Correct result:

b =  10 cm
c =  5 cm

#### Solution:

$V=250 \ \text{cm}^3 \ \\ S=250 \ \text{cm}^2 \ \\ a=5 \ \text{m} \ \\ V=abc \ \\ S=2(ab+bc+ac) \ \\ 50=bc \ \\ 125=5b + bc + 5c \ \\ c=50/b \ \\ 125=5b + 50 + 5 \cdot \ 50/b \ \\ 125b=5b^2 + 50b + 5*50 \ \\ \ \\ 125b=5b^2 + 50b + 5 \cdot \ 50 \ \\ -5b^2 +75b -250=0 \ \\ 5b^2 -75b +250=0 \ \\ \ \\ p=5; q=-75; r=250 \ \\ D=q^2 - 4pr=75^2 - 4\cdot 5 \cdot 250=625 \ \\ D>0 \ \\ \ \\ b_{1,2}=\dfrac{ -q \pm \sqrt{ D } }{ 2p }=\dfrac{ 75 \pm \sqrt{ 625 } }{ 10 } \ \\ b_{1,2}=\dfrac{ 75 \pm 25 }{ 10 } \ \\ b_{1,2}=7.5 \pm 2.5 \ \\ b_{1}=10 \ \\ b_{2}=5 \ \\ \ \\ \text{ Factored form of the equation: } \ \\ 5 (b -10) (b -5)=0 \ \\ \ \\ b=b_{1}=10 \ \text{cm}$

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$c=b_{2}=5 \ \text{cm}$

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