Cuboid Edges from Surface

The edges of a cuboid are in the ratio 1:2:3. Calculate their length if you know that the surface of the entire cuboid is S=5632 m². Then, perform a test to ensure the calculation is correct.

Final Answer:

a =  16 m
b =  32 m
c =  48 m

Step-by-step explanation:

$$\begin{gathered} S=5632{\text{m}}^2 \\ a=1x \\ b=2x \\ c=3x \\ S=2(ab+bc+ca) \\ S=2(1x\cdot 2x+2x\cdot 3x+3x\cdot 1x) \\ S=2(1\cdot 2+2\cdot 3+3\cdot 1)\cdot x^2 \\ r=2\cdot (1\cdot 2+2\cdot 3+3\cdot 1)=22 \\ S=r\cdot x^2 \\ x=\sqrt{\frac{S}{r}}=\sqrt{\frac{5632}{22}}=16\text{m} \\ a=1\cdot x=1\cdot 16=16\text{m} \end{gathered}$$

$b=2\cdot x=2\cdot 16=32\text{m}$

$$\begin{gathered} c=3\cdot x=3\cdot 16=48=48\text{m} \\ \text{ Verifying Solution: } \\ {S}_2=2\cdot (a\cdot b+b\cdot c+c\cdot a)=2\cdot (16\cdot 32+32\cdot 48+48\cdot 16)=5632{\text{m}}^2 \\ a:b:c=16:32:48=1:2:3 \end{gathered}$$

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