If the areas of adjacent faces of a cuboid (rectangular prism) are in the ratio of 2: 3: 5 and its volume is 900 cm³, then the length of the longest side is

A cuboid has dimensions $l$, $w$, and $h$. The three different face areas are:

• $l \times w$
• $w \times h$
• $l \times h$

The areas of adjacent faces are in the ratio $2 : 3 : 5$.

Using a variable $k$ to represent the ratio:

$l \times w = 2k$

$w \times h = 3k$

$l \times h = 5k$

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Multiplying all three equations:

$(l \times w) \times (w \times h) \times (l \times h) = 2k \times 3k \times 5k$

$l^2 \times w^2 \times h^2 = 30k^3$

$(l \times w \times h)^2 = 30k^3$

The volume is $l \times w \times h = 900$:

$(900)^2 = 30k^3$

$810000 = 30k^3$

$k^3 = 27000$

$k = 30$

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The actual face areas are:

$l \times w = 2(30) = 60$ cm²

$w \times h = 3(30) = 90$ cm²

$l \times h = 5(30) = 150$ cm²

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From $l \times w = 60$ and Volume $= 900$:

$h = \dfrac{900}{60} = 15$ cm

From $w \times h = 90$ and $h = 15$:

$w = \dfrac{90}{15} = 6$ cm

From $l \times w = 60$ and $w = 6$:

$l = \dfrac{60}{6} = 10$ cm

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The three dimensions are $10$ cm, $6$ cm, and $15$ cm.

Therefore, the length of the longest side is $15$ cm.