If the areas of adjacent faces of a cuboid (rectangular prism) are in…

The correct answer is Option (3) → 15 cm

If the sides of a cuboid are \(l\), \(b\) and \(h\), the areas of three adjacent faces are \(lb\), \(bh\) and \(hl\). Their product is \(l^2b^2h^2\), so the product of the three face areas equals the square of the volume.

The face areas (their product is the square of the volume):

• Let the face areas be \(lb=2k\), \(bh=3k\), \(hl=5k\)

• \(2k\times3k\times5k=(lbh)^2=900^2\)

• So \(30k^3=810000\)

• \(k^3=27000\), so \(k=30\)

• The face areas are 60, 90 and 150 cm²

The sides (each side is the volume divided by the face that does not contain it):

• \(l=\frac{900}{90}=10\)

• \(b=\frac{900}{150}=6\)

• \(h=\frac{900}{60}=15\), so the longest side is 15 cm