A cuboid has its length, width, and height in the ratio 6:5:4. If its total surface area measures 33,300 cm², what are the dimensions of the cuboid in centimeters?

Explanation

Let the common multiplier be x. Then, length = 6x, width = 5x, and height = 4x. The total surface area formula for a cuboid is 2(lw + wh + hl). Substituting the values, we get 2(6x × 5x + 5x × 4x + 6x × 4x) = 33,300. Simplifying, 2(30x² + 20x² + 24x²) = 33,300, which is 2(74x²) = 33,300, or 148x² = 33,300. Dividing both sides by 148 yields x² = 225, so x = 15. Therefore, the dimensions are length = 90 cm, width = 75 cm, and height = 60 cm.