Question

The dimensions of a cuboid are in the ratio 6:3:2. It’s volume is 2,304 $m^3$. find the dimensions and hence the total surface area of the cuboid.

Answer 1

Hint: We know that cuboid has three dimensions that is length, breadth, and height. Given that they are in the ratio 6:3:2. So we can take a variable and find all the dimensions in terms of that variable. We know the formula of volume which can be written in terms of that variable. Find the value of the variable using volume.

Complete step by step answer:
Let us suppose that
Length =l
Breadth=b
Height=h
Given that l:b:h = 6:3:2
=> l = 6a , b = 3a , h=2a
We know that the volume of the cuboid in terms of its length, breadth, height is lbh.
Given that volume is 2304$m^3$
=> lbh = 2304
By substituting l , b , h in terms of a in the above equation we get,
(6a) $ \times $ (3a) $ \times $ (2a) = 2304
=> 36$a^3$ = 2304
=> $a^3$ = 64
=> a = 4
=> l = 24m , b = 12m , h = 8m
We know that total surface area of a cuboid in terms of its length , breadth , height is 2(lb+bh+hl)
Which implies Total surface area = 2[ (24 $ \times $ 12) + (12 $ \times $ 8) + (8 $ \times $ 24) ] $m^2$
  = 2 [ 288 + 96 + 192 ] $m^2$
  = 2 [ 480 ] $m^2$ = 960 $m^2$
Therefore the dimensions are 24 , 12 , 8 and the total surface area of the given cuboid is 960 sq meter.

Note:
 Read all the geometric formulae regarding the lateral surface area, total surface area, volume, and dimensions of almost every known 2 dimensional or 3-dimensional body like square, rectangle, trapezium, parallelogram, rhombus, cube, cuboid, circle, sphere, cone, cylinder, tetrahedron, etc…