The length, breadth and height of a cuboidal box are in the ratio 7 : 5 : 3 and its whole surface area is 27832 $$cm^{2}$$. Its volume is:

Let the length, breadth and height of a cuboid box be 7x, 5x and 3x respectively.

Surface area = 27832 $$cm^{2}$$

2[(length $$\times$$ breadth) + (breadth $$\times$$ height) + (height $$\times$$ length)] = 27832

[(7x $$\times$$ 5x) + (5x $$\times$$ 3x) + (3x $$\times$$ 7x)] = 13916

$$35x^2 + 15x^2 + 21x^2 = 13916$$

$$71x^2 = 13916$$

$$x^2 = 13916/71 = 196$$

x = 14

Volume = length $$\times$$ breadth $$\times$$ height

= 7x $$\times$$ 5x $$\times$$ 3x = 105$$x^3$$

On putting the value of x,

= 105 $$\times (14^3) = 105 \times 2744 = 288120 cm^3$$