A cuboid has a surface area of 292 cm2. If the length and breadth of the…

Concept — a cuboid is bounded by three pairs of congruent rectangular faces, so its total surface area is TSA = 2(lb + bh + lh), where l, b and h are the length, breadth and height. Regrouping by dimension gives TSA = 2lb + 2h(l + b): the top-and-bottom pair is fixed as soon as the base is known, while the four side faces grow in direct proportion to the height, so the surface area is a linear function of h.

Application to this cuboid (l = 8 cm, b = 6 cm, TSA = 292 cm2):

1. Base pair: l × b = 8 × 6 = 48 cm2, so the top and bottom faces together contribute 2 × 48 = 96 cm2, independent of the height.

2. Side faces: the four vertical faces contribute 2h(l + b) = 2h(8 + 6) = 28h cm2.

3. Form the equation: 96 + 28h = 292.

4. Isolate the height term: 28h = 292 − 96 = 196.

5. Divide by the coefficient: h = 196 ÷ 28 = 7 cm.

Cross-check — substituting h = 7 cm into the unregrouped formula gives 2(8 × 6 + 6 × 7 + 8 × 7) = 2(48 + 42 + 56) = 2 × 146 = 292 cm2, which matches the given surface area. Because every extra centimetre of height adds exactly 28 cm2, the linear relation fixes one and only one height for a given surface area: 7.5 cm would produce 306 cm2 and 5 cm only 236 cm2.