Tag: volume of cube and cuboid

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Multiple choice maths how big? how heavy? measuring volume volume of solids volume of cube and cuboid

The radius of a cone is $\sqrt2$ times the height of the cone. A cube of maximum possible volume is cut from the same cone. What is the ratio of the volume of the cone to the volume of the cube?

  1. $3.18\pi$
  2. $2.25\pi$
  3. $2.35$

  4. Can't be determined
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Cube here will be inscribed in a cone as a square is in isosceles triangle.
Let the height of the cone be $h$
Radius=$\sqrt2 h$
Volume of cone=$\dfrac{1}{3}\pi r^2h$
                           =$\dfrac{2\sqrt 2}{3}\pi h^3$
Let the side of the cube be x,the top of the cone above it has the sign $(h-x)$ and radius $\dfrac{x}{2}$
Using properties of similar triangle $\dfrac { \dfrac { x }{ 2 }  }{ h-x } =\dfrac{\sqrt2 h}{h}$
                                                            $=\sqrt 2 x$
                                                             $=\dfrac { 2\sqrt { 2 } h }{ 2\sqrt { 2 } +1 } $
Volume of the cube=$\dfrac { 2\sqrt { 2 } h }{ 2\sqrt { 2 } +1 } $
Ratio of the volume of the cone to volume of the cube=$\dfrac { \dfrac { 2\sqrt { 2 }  }{ 3 } \pi h^{ 3 } }{ (\dfrac { 2\sqrt { 2 } h }{ 2\sqrt { 2 } +1 } )^ 3 } $
                                            $=\dfrac{\pi(2\sqrt { 2 } +1  )^ 3)}{24}$
                                            $=2.35\pi$

Multiple choice maths how big? how heavy? measuring volume volume of solids volume of cube and cuboid

If a box is $\dfrac{1}{4}$ filled contains $5$ small cubes each of volume $1$ cubic units then find out the volume of the box.

  1. $25$ cu.
  2. $20$ cu.
  3. $15$ cu.
  4. $5$ cu.
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Since the box is $\dfrac {1}{4}$ filled with  the given cubes , we need to multiply the total volume of the given cubes by $4$ to get the total volume of thr box.

Volume of one cube is $1 cu.$
$\therefore $ Volume of 5 cubes will be $5\times 1 cu.=5cu.$
$\therefore $ Volume of the box will be $4\times 5 cu.=20cu.$

Multiple choice maths how big? how heavy? measuring volume volume of solids volume of cube and cuboid

If the volumes of two cubes are in the ratio $8:1$, then the ratio of their edges is

  1. $8:1$
  2. $2\sqrt 2:1$
  3. $2:1$

  4. none of these
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Let $V _1$ and $V _2$ be two volume of cubes.

$l _1$ and $l _2$ be edges of the two cubes.
We know that,
Volume of cube $V=l^3$
So,
$\Rightarrow$  $\dfrac{V _1}{V _2}=\dfrac{l _1^3}{l _2^3}$

$\Rightarrow$  $\dfrac{8}{1}=\left(\dfrac{l _1}{l _2}\right)^3$             [ Given ]

$\therefore$  $\dfrac{l _1}{l _2}=\dfrac{2}{1}$

$\therefore$  Ratio of their edges is $2:1$.

Multiple choice maths how big? how heavy? measuring volume volume of solids volume of cube and cuboid

The volume of a cube whose surface area is $96{cm}^{2}$, is

  1. $16\sqrt 2{cm}^{3}$
  2. $32{cm}^{3}$
  3. $64{cm}^{3}$
  4. $216{cm}^{3}$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Let $l$be the side of cube.

Surface area of cube $=6l^2$
$\Rightarrow$  $96=6l^2$                      [ Given ]
$\Rightarrow$  $l^2=16$
$\therefore$  $l=4\,cm$
Now,
$\Rightarrow$  Volume of cube $=l^3$
                                  $=(4)^3$
                                  $=64\,cm^3$

Multiple choice maths how big? how heavy? measuring volume volume of solids volume of cube and cuboid

If each edge of a cube, of volume $V$, is doubled, then the volume of the new cube is

  1. $2V$
  2. $4V$
  3. $6V$
  4. $8V$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Let $a$ be the initial edge of the cube.

So, 
Volume of cube $V=a^3$
In the new cube,
Let $a'$ be the edge of new cube
$\therefore$  $a'=2a$               [ Given ]
Volume of new cube,
$V'=(a')^3$
      $=(2a)^3$
      $=8a^3$
      $=8V$                        [ Since, $a^3=V$ ]
Volume of the new cube is $8V.$