# 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 length 2 cm. Find the length of a diagonal of the cuboid. Give your answer correct to 3 sig. Fig.

Result

d =  13.748 cm

#### Solution:

$V=40 \ \text{cm}^3 \ \\ S=100 \ \text{cm}^2 \ \\ a=2 \ \text{cm} \ \\ V=abc \ \\ S=2(ab+bc+ac) \ \\ \ \\ 20=bc \ \\ 50=2b + bc + 2c \ \\ \ \\ 20/b=c \ \\ \ \\ 50b=2b^2 + 20b + 2 \cdot \ 20 \ \\ -2b^2 +30b -40=0 \ \\ 2b^2 -30b +40=0 \ \\ \ \\ p=2; q=-30; r=40 \ \\ D=q^2 - 4pr=30^2 - 4\cdot 2 \cdot 40=580 \ \\ D>0 \ \\ \ \\ b_{1,2}=\dfrac{ -q \pm \sqrt{ D } }{ 2p }=\dfrac{ 30 \pm \sqrt{ 580 } }{ 4 }=\dfrac{ 30 \pm 2 \sqrt{ 145 } }{ 4 } \ \\ b_{1,2}=7.5 \pm 6.0207972893961 \ \\ b_{1}=13.520797289396 \ \\ b_{2}=1.4792027106039 \ \\ \ \\ \text{ Factored form of the equation: } \ \\ 2 (b -13.520797289396) (b -1.4792027106039)=0 \ \\ \ \\ b=b_{1}=13.5208 \doteq 13.5208 \ \text{cm} \ \\ c=20/b=20/13.5208 \doteq 1.4792 \ \text{cm} \ \\ \ \\ V_{1}=a \cdot \ b \cdot \ c=2 \cdot \ 13.5208 \cdot \ 1.4792=40 \ \text{cm}^3 \ \\ S_{1}=2 \cdot \ (a \cdot \ b+b \cdot \ c+a \cdot \ c)=2 \cdot \ (2 \cdot \ 13.5208+13.5208 \cdot \ 1.4792+2 \cdot \ 1.4792)=100 \ \text{cm}^2 \ \\ \ \\ d=\sqrt{ a^2+b^2+c^2 }=\sqrt{ 2^2+13.5208^2+1.4792^2 } \doteq 3 \ \sqrt{ 21 } \doteq 13.7477 \doteq 13.748 \ \text{cm}$

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