Tag: surface area and volume of sphere

Questions Related to surface area and volume of sphere

Consider an incomplete pyramid of balls on a square base having $18$ players, and having $13$ balls on each side of the top layer. Then the total number $N$ of balls in that pyramid satisfies 

maths measures and the circle surface area and volume of sphere surface area of a prism surface area of a prism and a pyramid

  1. $ 9000 < N <10000$

  2. $8000 < N < 9000$

  3. $7000 < N < 8000$

  4. $ 10000 < N < 12000 $

Show answer
Correct Option: B
Explanation:

Top layer has $(13 \times 13)$ balls 
Similarly one layer below top layer will have $(14 \times 14) $ balls and we have $18$ lesens to total number of ball 
$ N = (13)^2 + (14)^2 + . . . . + (30)^2 $

$\displaystyle N = \frac {30 \times 31 \times 61} {6} = \frac {12 \times 13 \times 25} {6}$

$ N = 8805 $

The circumference of a 1 cm thick pipe is 44 cm. The level of water that 7 cm of pipe can hold is

maths measures and the circle surface area and volume of sphere surface area of a prism surface area of a prism and a pyramid

  1. $798 cm^3$

  2. $308 cm^3$

  3. $792 cm^3$

  4. $795 cm^3$

Show answer
Correct Option: C
Explanation:

Given that

$2\pi r=44$
$r=7\ cm$

So, the inner radius of the pipe $=7-1=6\ cm$

Therefore, the volume of the pipe
$=\pi r^2h$
$=\pi \times 6^2\times 7$
$=792\ cm^3$

Hence, this is the answer.

The base of a right prism is a square of perimeter 20 cm and its height is 30 cm. The volume of the prism is

maths measures and the circle surface area and volume of sphere surface area of a prism surface area of a prism and a pyramid

  1. $700 cm^3$

  2. $750 cm^3$

  3. $800 cm^3$

  4. $850 cm^3$

Show answer
Correct Option: B
Explanation:

Given, perimeter $=4a=20$cm
$\therefore a=5$ cm
Area $=a^2=25  cm^2$
Volume $=$ Area $\times$ Height
Volume $=25 \times 30$
Volume $=750  cm^3$

The base of a right prism is an equilateral triangle of edge $12$m. If the volume of the prism is $288\sqrt 3m^3$, then its height is:

maths measures and the circle surface area and volume of sphere surface area of a prism surface area of a prism and a pyramid

  1. $6$m

  2. $8$m

  3. $10$m

  4. $12$m

Show answer
Correct Option: B
Explanation:

length of Equilateral triangle $= 12 m$
Area of equilateral triangle = $\displaystyle \frac{\sqrt{3}}{4}a^2$ = $\displaystyle \frac{\sqrt{3}}{4}(12)^2$ = $36\sqrt{3}$
Volume of prism = $288\sqrt{3} m^3$ = Area of triangle X height
$288\sqrt{3} m^3$ = $36 \sqrt{3} \times$ height
$\therefore $ Height $= 8 m$

If the circumference of the inner edge of a hemispherical bowl is $\displaystyle\frac{132}{7}:cm$, then what is the capacity?

maths solids volume of a sphere surface area and volume of sphere surface areas and volumes surface areas and volumes of solids problems involving volume of combined solids application of surface area and volume of solids

  1. $12\pi:cm^3$

  2. $18\pi:cm^3$

  3. $24\pi:cm^3$

  4. $36\pi:cm^3$

Show answer
Correct Option: B
Explanation:

$\displaystyle2\pi r=\frac{132}{7}\implies r=3$

$\therefore$ Capacity $\displaystyle=\frac{2}{3}\pi r^3=18\pi$

The side of a cube is equal to diameter of the sphere. The ratio of volumes 
of cube and sphere is

maths solids volume of a sphere surface area and volume of sphere surface areas and volumes surface areas and volumes of solids problems involving volume of combined solids application of surface area and volume of solids

  1. $\frac{11}{12}$

  2. $\frac{22}{11}$

  3. $\frac{11}{21}$

  4. $\frac{21}{11}$

Show answer
Correct Option: D
Explanation:

Let the sides of cube be $s$ and radius of sphere be $r$

then $s=2r=(2r)^3=8r^3$
Volume of a cube$=s^3$
Volume of a sphere$=\cfrac{4}{3}\pi r^3$
Ratio of volume of cube and sphere$=\cfrac{8r^3}{\cfrac{4}{3}\pi r^3}$
$=\cfrac{8r^3}{\cfrac{4}{3}\times \cfrac{22}{7} r^3}\=\cfrac{21}{11}$

A hollow spherical shell is made of metal of density $4.8$ g/cm$^3$. If its internal and external radii are $10$ cm and $12$ cm respectively, find the weight of the shell

maths solids volume of a sphere surface area and volume of sphere surface areas and volumes surface areas and volumes of solids problems involving volume of combined solids application of surface area and volume of solids

  1. $15.24 $ kg

  2. $12.84 $ kg

  3. $14.64 $ kg

  4. None of these

Show answer
Correct Option: C
Explanation:

Volume of spherical shell
$= \displaystyle \frac{4 \pi}{3} (R^3 - r^3) = \frac{4 \pi}{3} (12^3 - 10^3)$
$=\displaystyle \frac{4}{3} \times \pi \times (12 -10) (12^2 +12 \times 10 +10^2)$
$=\displaystyle \frac{4}{3} \times \pi \times 2 \times 264 cm^3$
$Weight = volume \times density = \displaystyle \frac{4}{3}\times \pi \times 364 \times 4.8 = 14.64 kg$

The radius of a sphere is r and radius of base of a cylinder is r and height is 2r. The ratio of their volumes will be-

maths solids volume of a sphere surface area and volume of sphere surface areas and volumes surface areas and volumes of solids problems involving volume of combined solids application of surface area and volume of solids

  1. $2:3$

  2. $3:4$

  3. $4:3$

  4. $3:2$

Show answer
Correct Option: A
Explanation:

Given,

The radius of the sphere $=r$

The radius of the cylinder $=r$

The height of the cylinder $=2r$

 

We know that the volume of the sphere

${{V} _{1}}=\dfrac{4}{3}\pi {{r}^{3}}$

 

We know that the volume of the cylinder

$ {{V} _{2}}=\pi {{r}^{2}}h $

$ {{V} _{2}}=\pi {{r}^{2}}\left( 2r \right) $

$ {{V} _{2}}=2\pi {{r}^{3}} $

 

Therefore, the required ratio

$ \dfrac{{{V} _{1}}}{{{V} _{2}}}=\dfrac{\dfrac{4}{3}\pi {{r}^{3}}}{2\pi {{r}^{3}}} $

$ \dfrac{{{V} _{1}}}{{{V} _{2}}}=\dfrac{2\pi {{r}^{3}}}{3\pi {{r}^{3}}} $

$ \dfrac{{{V} _{1}}}{{{V} _{2}}}=\dfrac{2}{3} $

$ {{V} _{1}}:{{V} _{2}}=2:3 $

 

Hence, this is the answer.

The volume of a spherical shell whose internal and external diameters are $8cm$ and $10cm$ respectively (in cubic cm) is:

maths solids volume of a sphere surface area and volume of sphere surface areas and volumes surface areas and volumes of solids problems involving volume of combined solids application of surface area and volume of solids

  1. $\cfrac{122\pi}{3}$

  2. $\cfrac{244\pi}{3}$

  3. $212$

  4. $257$

Show answer
Correct Option: B
Explanation:

Given, internal diameter $=8cm$ and external diameter $=10cm$
Volume of a hollow sphere of outer Radius R and inner radius r $ = \frac { 4 }{ 3 } \pi ({R}^{2} -{ r}^{ 3 }) $
Inner radius of the spherical shell $ = \frac {8}{2} = 4 cm $
Outer radius of the spherical shell $ = \frac {10}{2} = 5  cm $
Hence, volume of spherical shell $ = \frac { 4 }{ 3 } \times \pi \times ({5}^{3} - {4}^{3}) = \frac { 4 }{ 3 } \times \pi  \times 61 = \frac {244\pi}{3}  { cm }^{ 3 }  $

A metallic hemispherical bowl is $0.25\;cm$ thick. The inside radius of the bowl is $5\;cm$. Find the volume of steel used in making the bowl.

maths solids volume of a sphere surface area and volume of sphere surface areas and volumes surface areas and volumes of solids problems involving volume of combined solids application of surface area and volume of solids

  1. $43.25\;cm^3$

  2. $41.27\;cm^3$

  3. $42.25\;cm^3$

  4. $40.25\;cm^3$

Show answer
Correct Option: B
Explanation:

Volume of speed used $=$ $\dfrac { 2 }{ 3 } \pi \left( { r } _{ 1 }^{ 3 }-{ r } _{ 2 }^{ 3 } \right) $

${ r } _{ 1 }=5+0.25=5.25cm$
${ r } _{ 2 }=5cm$
$\therefore \quad $ Volume $=$ $\dfrac { 2 }{ 3 } \times \dfrac { 22 }{ 7 } \times \left( { \left( 5.25 \right)  }^{ 3 }-{ \left( 5 \right)  }^{ 3 } \right) $

                       $= \dfrac { 44 }{ 21 } \times \left( 144.70-125 \right) $

                       $= 41.27$ ${ cm }^{ 3 }$