Question

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

i) If its surface area is 2548 $c{m}^2$, find its volume

ii) If its volume is $3000m^3$, find its surface area.

Answer 1

Surface area of cuboid $=2548c{m}^2$
i) Ratio in length, breadth and height of a cuboid
= 4 : 3 : 2
Let length = 4x
Breadth = 3x
and height = 2x
$\therefore$ Surface area $=2(4x\times 3x+3x\times 2x+2x\times 4x)=2(12x^2+6x^2+8x^2)=2\times 26x^2=52x^2\therefore 52x^2=2548x^2=\frac{2548}{52}=49=(7{)}^2\therefore x=7$
$\therefore$ Length $=4x=4\times 7=28cm$
Breadth $=3x=3\times 7=21cm$
and height $=2x=2\times 7=14cm$
$\therefore$ Volume = lbh
$=28\times 21\times 14c{m}^3=8232c{m}^2$

ii) If volume $=3000m^3$
$\Rightarrow 4x\times 3x\times 2x=3000\Rightarrow 24x^3=3000\Rightarrow x^3=\frac{3000}{24}=125=(5{)}^3\therefore x=5m$
Length $=5\times 4=20$, breadth $=5\times 3=15m$ and height $=5\times 2=10m$
$\therefore$ Surface area =2[lb+bh+hl]
$=2[20\times 15+15\times 10+10\times 20]m^2$
$=2[300+150+200]m^2=2\times 650=1300m^2$