Cuboid diagonal

Calculate the volume and surface area of the cuboid ABCDEFGH, which sides abc has dimensions in the ratio of 9:3:8 and if you know that the wall diagonal AC is 86 cm and angle between AC and the body diagonal AG is 25 degrees.

Correct result:

a =  45.12 cm
b =  15.04 cm
c =  40.1 cm
V =  27208 cm3
S =  6182 cm2

Solution:

a=86⋅ tan⁡25∘/8⋅ 9=tan⁡5π/36=0=45.12 cma=86 \cdot \ \tan 25 ^\circ / 8 \cdot \ 9=\tan 5π/36=0=45.12 \ \text{cm}
b=86⋅ tan⁡25∘/8⋅ 3=tan⁡5π/36=0=15.04 cmb=86 \cdot \ \tan 25 ^\circ / 8 \cdot \ 3=\tan 5π/36=0=15.04 \ \text{cm}
tan⁡25∘=c∣AC∣ c=∣AC∣tan⁡25∘=40.1 cm a=98c=45.12 b=38c=15.04 cm\tan 25 ^\circ = \dfrac{c}{|AC|} \ \\ c = |AC| \tan 25 ^\circ = 40.1 \ \text{cm} \ \\ a = \dfrac{ 9 }{ 8 } c = 45.12 \ \\ b = \dfrac{ 3 }{ 8 } c = 15.04 \ \text{cm}
V=abc=27208 cm3 V = abc = 27208 \ \text{cm}^3
S=2(ab+bc+ac)=6182 cm2 S = 2(ab+bc+ac) = 6182 \ \text{cm}^2

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