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:

$$\begin{gathered} V=250{\text{cm}}^3 \\ S=250{\text{cm}}^2 \\ a=5\text{ m } \\ V=abc \\ S=2(ab+bc+ac) \\ 50=bc \\ 125=5b+bc+5c \\ c=50\mathrm{/}b \\ 125=5b+50+5\cdot 50\mathrm{/}b \\ 125b=5b^2+50b+5\ast 50 \\ 125b=5b^2+50b+5\cdot 50 \\ -5b^2+75b-250=0 \\ 5b^2-75b+250=0 \\ p=5;q=-75;r=250 \\ D=q^2-4pr=75^2-4\cdot 5\cdot 250=625 \\ D>0 \\ {b}_{1,2}={\displaystyle \frac{-q\pm \sqrt{D}}{2p}}={\displaystyle \frac{75\pm \sqrt{625}}{10}} \\ {b}_{1,2}={\displaystyle \frac{75\pm 25}{10}} \\ {b}_{1,2}=7.5\pm 2.5 \\ {b}_1=10 \\ {b}_2=5 \\ \text{ Factored form of the equation: } \\ 5(b-10)(b-5)=0 \\ b=b_1=10=10\text{ cm} \end{gathered}$$

Our quadratic equation calculator calculates it.

$c=b_2=5=5\text{ cm}$

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