The length, breadth, and height of a closed cardboard box are in the ratio of 5 : 3 : 7 respectively. Find the length of this box, if its total surface area is 1278 $$cm^{2}$$.

The length, breadth, and height of a closed cardboard box are in the ratio of 5 : 3 : 7 respectively.

Let's assume the length, breadth, and height of a closed cardboard box are 5y, 3y and 7y respectively.

Total surface area is 1278 $$cm^{2}$$.

Total surface area = $$2(length \times breadth + breadth \times height + height \times length)$$

$$1278=2(5y\times3y+3y\times7y+7y\times5y)$$

$$639=(15y^2+21y^2+35y^2)$$

$$639=71y^2$$

so length of this box = 5y

= $$5\times3$$

= 15 cm