Cuboid

Cuboid with edge a=6 cm and space 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:

$$\begin{gathered} a=6\text{ cm } \\ V=900{\text{cm}}^3 \\ u=31\text{ cm } \\ V=abc \\ 900=6bc \\ bc=150 \\ u=\sqrt{a^2+b^2+c^2} \\ 31^2=6^2+b^2+c^2 \\ 925=b^2+c^2 \\ 925={\displaystyle \frac{22500}{c^2}}+c^2 \\ {c}^4-925c^2+22500=0 \\ x=c^2 \\ {x}^2-925x+22500=0 \\ {x}^2-925x+22500=0 \\ a=1;b=-925;c=22500 \\ D=b^2-4ac=925^2-4\cdot 1\cdot 22500=765625 \\ D>0 \\ {x}_{1,2}={\displaystyle \frac{-b\pm \sqrt{D}}{2a}}={\displaystyle \frac{925\pm \sqrt{765625}}{2}} \\ {x}_{1,2}={\displaystyle \frac{925\pm 875}{2}} \\ {x}_{1,2}=462.5\pm 437.5 \\ {x}_1=900 \\ {x}_2=25 \\ \text{ Factored form of the equation: } \\ (x-900)(x-25)=0 \\ {c}_1=\sqrt{x_1}=\sqrt{900}=30 \\ {c}_2=\sqrt{x_2}=\sqrt{25}=5 \\ b=c_1=30=30\text{ cm} \end{gathered}$$

Our quadratic equation calculator calculates it.

$$\begin{gathered} c=c_2=5=5\text{ cm } \\ \text{ Verifying Solution: } \\ d=\sqrt{a^2+b^2+c^2}=\sqrt{6^2+30^2+5^2}=31 \\ d=u \\ {V}_2=a\cdot b\cdot c=6\cdot 30\cdot 5=900{\text{cm}}^3 \\ {V}_2=V \end{gathered}$$

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