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:

$a=86 \cdot \ \tan 25 ^\circ / 8 \cdot \ 9=\tan 5π/36=0=45.12 \ \text{cm}$
$b=86 \cdot \ \tan 25 ^\circ / 8 \cdot \ 3=\tan 5π/36=0=15.04 \ \text{cm}$
$\tan 25 ^\circ = \dfrac{c}{|AC|} \ \\ c = |AC| \tan 25 ^\circ = 40.1 \ \text{cm} \ \\ a = \dfrac{ 9 }{ 8 } c = 45.12 \ \\ b = \dfrac{ 3 }{ 8 } c = 15.04 \ \text{cm}$
$V = abc = 27208 \ \text{cm}^3$
$S = 2(ab+bc+ac) = 6182 \ \text{cm}^2$

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