length, breadth and height of a cuboid are in the ratio

Solution

Let the dimensions be 6x, 4x, and 3x. The formula for the total surface area (TSA) of a cuboid is: TSA = 2(lb+bh+hl) Substituting the dimensions: TSA = 2((6x×4x) + (4x×3x) +(3x×6x)) =2(24x² + 12x² + 18x²) = 2(54x²) = 108x² Given the TSA is 972 cm²: 108x²= 972 x² = 972/108 x² = 9 x=3 now the dimensions of the cuboid- length =6x=6×3=18cm breadth =4x =4×3=12cm height =3x=3×3=9cm The length of the diagonal d of a cuboid is given by- d² =1²+b²+ h² Substituting the values: d²=18²+12² +9² d²=324+ 144 +81 d = √549 d=3√61