Solid cuboid

A solid cuboid has a volume of 40 cm³. The cuboid has a total surface area of 100 cm squared. One edge of the cuboid has a length of 2 cm. Find the length of a diagonal of the cuboid. Give your answer correct to 3 sig. Fig.

Correct answer:

d =  13.7477 cm

Step-by-step explanation:

$$\begin{gathered} V=40{\text{cm}}^3 \\ S=100{\text{cm}}^2 \\ a=2\text{cm} \\ V=abc \\ S=2(ab+bc+ac) \\ 20=bc \\ 50=2b+bc+2c \\ 20/b=c \\ 50b=2b^2+20b+2\cdot 20 \\ 50b=2b^2+20b+2\cdot 20 \\ -2b^2+30b-40=0 \\ 2b^2-30b+40=0 \\ 2...\text{ prime number} \\ 30=2\cdot 3\cdot 5 \\ 40=2^3\cdot 5 \\ \text{GCD}(2,30,40)=2 \\ {b}^2-15b+20=0 \\ p=1;q=-15;r=20 \\ D=q^2-4pr=15^2-4\cdot 1\cdot 20=145 \\ D>0 \\ {b}_{1,2}=\frac{-q\pm \sqrt{D}}{2p}=\frac{15\pm \sqrt{145}}{2} \\ {b}_{1,2}=7.5\pm 6.020797 \\ {b}_1=13.520797289 \\ {b}_2=1.479202711 \\ b=b_1=13.5208\doteq 13.5208\text{cm} \\ c=20/b=20/13.5208\doteq 1.4792\text{cm} \\ {V}_1=a\cdot b\cdot c=2\cdot 13.5208\cdot 1.4792=40{\text{cm}}^3 \\ {S}_1=2\cdot (a\cdot b+b\cdot c+a\cdot c)=2\cdot (2\cdot 13.5208+13.5208\cdot 1.4792+2\cdot 1.4792)=100{\text{cm}}^2 \\ d=\sqrt{a^2+b^2+c^2}=\sqrt{2^2+13.5208^2+1.4792^2}=3\sqrt{21}=13.7477\text{cm} \end{gathered}$$

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