The length, breadth, and height of a cuboid (rectangular solid) are 4 : 3: 2.

(i) If its surface area is 2548 cm^{2}, find its volume.

(ii) If its volume is 3000 m^{3}, find its surface area.

#### Solution

Surface area of cuboid = 2548 cm^{2}

Ratio in length, breadth and height of a cuboid = 4 : 3 : 2

Let length = 4x, Breadth = 3x and height = 2x

`therefore "Surface area" = 2(4x xx 3x + 3x xx 2x + 2x xx 4x)`

= `2(12x^2 + 6x^2 + 8x^2)`

= `2 xx 26x^2 = 52x^2`

`therefore 52x^2 = 2548`

`x^2 = 2548/52 = 49 = (7)^2`

`therefore x = 7`

`therefore "Length" = 4x = 4 xx 7 = 28` cm

`therefore "Breadth" = 3x = 3 xx 7 = 21` cm

`"and height" = 2x = 2 xx 7 = 14`cm

`therefore "Volume" = lbh`

`= 28 xx 21 xx 14` cm^{3 }= 8232 cm^{2}

(ii) If volume = 3000 m^{3}

⇒ `4x xx 3x xx 2x = 3000`

⇒ `24x^3 = 3000`

⇒ `x^3 = 3000/24 = 125 = (5)^3`

`therefore x = 5`m

`"Length" = 5 xx 4 = 20, "breadth" = 5 xx 3 = 15`m

and height = `5 xx 2 = 10`m

`therefore "Surface area" = 2[lb + bh + hl]`

= `2[20 xx 15 + 15 xx 10 + 10 xx 20]`m^{2}

= `2[300 + 150 + 200]`m^{2}

= `2 xx 650 = 1300`m^{2}