Tag: measuring volume

Questions Related to measuring volume

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?

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

  1. $3.18\pi$

  2. $2.25\pi$

  3. $2.35$

  4. Can't be determined

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Correct Option: B
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 } $
                                            $=\dfrac{\pi(2\sqrt { 2 } +1  )^ 3)}{24}$
                                            $=2.35\pi$

If S is the total surface area of a cube and V is its volume, then which one of the following is correct?

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

  1. $V^{3} = 216 S^{2}$

  2. $S^{3} = 216 V^{2}$

  3. $S^{3} = 6 V^{2}$

  4. $S^{2} = 36 V^{3}$

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Correct Option: B
Explanation:

$S = 6a^{2}; V = a^{3}$
Then $S^{3} = 216a^{6} = 216(a^{3})^{2}$ or $S^{3} = 216 V^{2}$

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.

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

  1. $25$ cu.

  2. $20$ cu.

  3. $15$ cu.

  4. $5$ cu.

Show answer
Correct Option: B
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 $4\times 5 cu.=20cu.$

How many small cubical blocks side $5$cm can be cut from a cubical block whose each edge measure $20$cm?

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

  1. $56$

  2. $48$

  3. $64$

  4. $52$

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Correct Option: A

how many bricks each measuring $250 cm$ by $12.5 cm$ by $7.5 cm$ will be required to build a wall 5 m long ,3m high and 20 m thick?

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

  1. $148$

  2. $128$

  3. $168$

  4. $158$

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Correct Option: A

how many bricks are required to build a wall 15 m long 3 m high and 50 cm thick ,if each brick measures 25 cm by 12 cm by 6 cm?

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

  1. $16500$

  2. $14500$

  3. $12500$

  4. $10500$

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Correct Option: A

How many cubes each of surface area $24 sq\ m$ can be made out a meter cube, without any wastage?

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

  1. $75$

  2. $250$

  3. $125$

  4. $62$`

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Correct Option: A

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

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

  1. $8:1$

  2. $2\sqrt 2:1$

  3. $2:1$

  4. none of these

Show answer
Correct Option: C
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 $2:1$.

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

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

  1. $16\sqrt 2{cm}^{3}$

  2. $32{cm}^{3}$

  3. $64{cm}^{3}$

  4. $216{cm}^{3}$

Show answer
Correct Option: C
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 $=l^3$
                                  $=(4)^3$
                                  $=64\,cm^3$

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

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

  1. $2V$

  2. $4V$

  3. $6V$

  4. $8V$

Show answer
Correct Option: D
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 $8V.$