Tag: solids

Questions Related to solids

Find the surface area of a cone whose radius is $20$ cm and the slant height is $0.3$ cm. (use $\displaystyle \pi =3.14$)

maths solids surface area of a cone surface area of cone the surface area of a cone

  1. $\displaystyle 1274.84\ m$

  2. $\displaystyle 1274.84{ \ mm }^{ 2 }$

  3. $\displaystyle 1274.84{ \  cm }^{ 2 }$

  4. $\displaystyle 1274.84\ cm$

Show answer
Correct Option: C
Explanation:

Given, radius $=20$ cm and height $=0.$ cm

Surface area of a cone $=$ $\displaystyle \pi r\left( r+s \right) $
$\displaystyle =3.14\times 20\times \left( 20+0.3 \right) $
$\displaystyle =3.14\times 20\times 20.3$
$\displaystyle =1274.84{ cm }^{ 2 }$

The curved surface area of a cone is $\displaystyle 320{\  m }^{ 2 }$ whose radius is $7$ m. Find the surface area of a cone. (Assume $\displaystyle \pi =\frac { 22 }{ 7 } $)

maths solids surface area of a cone surface area of cone the surface area of a cone

  1. $\displaystyle 474{\  m }^{ 2 }$

  2. $\displaystyle 424{\  m }^{ 2 }$

  3. $\displaystyle 404{ \ m }^{ 2 }$

  4. $\displaystyle 454\ m$

Show answer
Correct Option: A
Explanation:

Given, surface area of cone $=20$ $m^2$ and radius $7$ m

Curved surface area of a cone $=$ $\displaystyle \pi rs$
$\displaystyle =320+\left( { 22 }/{ 7 } \right) \times 7\times 7$
$\displaystyle =320+154$
$\displaystyle =474{ m }^{ 2 }$

Calculate the surface area of a cone whose radius is $\dfrac{1}{3}$ cm and slant height is $12$ cm.

maths solids surface area of a cone surface area of cone the surface area of a cone

  1. ${3\pi}$ $cm^2$

  2. $\dfrac { 37\pi }{ 9 }$ $ { cm }^{ 2 }$

  3. $\dfrac { 37\pi }{ 8 }$ $ { cm }^{ 2 }$

  4. $\dfrac { 5\pi }{ 9 }$ $ { cm }^{ 2 }$

Show answer
Correct Option: B
Explanation:

Surface area of cone is $A=πr(r+l)$


Here, radius is $r=\dfrac { 1 }{ 3 }$ cm and slant height is $l=12$ cm.

Thus,
 
$A=πr(r+l)=π\times \dfrac { 1 }{ 3 } \left( \dfrac { 1 }{ 3 } +12 \right) =\dfrac { 1 }{ 3 } π\times \dfrac { 37 }{ 3 } =\dfrac { 37π }{ 9 }$ cm$^2$
 
Hence, the surface area of the cone is $\dfrac { 37π }{ 9 }$ cm$^2$.

A cone has a radius of $2$ cm and height of $3$ cm, find total surface area of the cone.

maths solids surface area of a cone surface area of cone the surface area of a cone

  1. $\displaystyle 35.168\ cm$

  2. $\displaystyle 36.158{ \ cm }^{ 2 }$

  3. $\displaystyle 35.168{  cm }^{ 2 }$

  4. $\displaystyle 35.168{\  m }^{ 2 }$

Show answer
Correct Option: C
Explanation:

First need to find the value of slant height(s) of a cone, using Pythagoras theorem, since the cross section is a right triangle.
$\displaystyle { s }^{ 2 }={ h }^{ 2 }+{ r }^{ 2 }$
$\displaystyle { s }^{ 2 }={ 3 }^{ 2 }+{ 2 }^{ 2 }$
$\displaystyle { s }^{ 2 }=9+4$
$\displaystyle s=\sqrt { \left( 13 \right)  } $
$\displaystyle s=3.6$ cm
Then, surface area of a cone $\displaystyle =\pi rs+\pi { r }^{ 2 }$
$\displaystyle =3.14\times 2\times 3.6+3.14\times 2\times 2$
$\displaystyle =22.608+12.56$
$\displaystyle= 35.168{ cm }^{ 2 }$

A conical water tank has a radius of $0.2$ mm and height of $1.2$ mm, find total surface area of the tank.

maths solids surface area of a cone surface area of cone the surface area of a cone

  1. $\displaystyle 0.9734\ cm$

  2. $\displaystyle 0.9734{ \ cm }^{ 2 }$

  3. $\displaystyle 0.9734\ mm$

  4. $\displaystyle 0.9734{\  mm }^{ 2 }$

Show answer
Correct Option: D
Explanation:

Given, radius $=0.2$ mm and height $=1.2$ mm 
Then find the value of slant height(s) of a conical water tank, using Pythagoras theorem, since the cross section is a right triangle.
$\displaystyle { s }^{ 2 }={ h }^{ 2 }+{ r }^{ 2 }$
$\displaystyle { s }^{ 2 }={ 1.2 }^{ 2 }+{ 0.2 }^{ 2 }$
$\displaystyle { s }^{ 2 }=1.44+0.4$
$\displaystyle s=\sqrt { \left( 1.84 \right)  } $
$\displaystyle s=1.35$ mm
So, surface area of a cone $\displaystyle =\pi rs+\pi { r }^{ 2 }$
$\displaystyle =3.14\times 0.2\times 1.35+3.14\times 0.2\times 0.2$
$\displaystyle =0.8478+0.1256$
$\displaystyle =0.9734{\  mm }^{ 2 }$

Calculate the surface area of cone whose curved surface area is $\displaystyle 100{ \ cm }^{ 2 }$ and its radius $200$ cm.

maths solids surface area of a cone surface area of cone the surface area of a cone

  1. $\displaystyle 125700\ cm$

  2. $\displaystyle 125700{\  mm }^{ 2 }$

  3. $\displaystyle 125700{\  m }^{ 2 }$

  4. $\displaystyle 125700{\  cm }^{ 2 }$

Show answer
Correct Option: D
Explanation:

Surface area of a cone $= $ Curved surface area of a cone $+$ Area of circle
So, SA $\displaystyle =\pi rs+\pi { r }^{ 2 }$
$\displaystyle =100+3.14\times 200\times 200$
$\displaystyle =100+125600$
$\displaystyle =125700{ cm }^{ 2 }$

Find the surface area of a conical hat. If its slant height is three times the radius, the base diameter of a hat is $4$ inches. (Use $\displaystyle \pi =3$)

maths solids surface area of a cone surface area of cone the surface area of a cone

  1. $\displaystyle 48{ \ in }^{ 2 }$

  2. $\displaystyle 36\ in$

  3. $\displaystyle 48\ in$

  4. $\displaystyle 46{\  in }^{ 2 }$

Show answer
Correct Option: A
Explanation:

Surface area of cone is $A=πr(r+l)$


Here, the diameter is $4$ in and therefore, the radius is half of diameter that is $r=2$ in and it is also given that slant height is thrice the radius that is $l=(3\times 2)=6$ in. We use $π=3$.

Thus,
 
$A=πr(r+l)=3\times 2\left( 2+6 \right) =3\times 2\times 8=48$ in$^2$
 
Hence, the surface area of the conical hat is $48$ in$^2$.

Find the total surface area of a cone, if its radius $14$ m and slant height $49$ m. (Use $\displaystyle \pi =\frac { 22 }{ 7 } $).

maths solids surface area of a cone surface area of cone the surface area of a cone

  1. $\displaystyle 2762$ $\ m^2$

  2. $\displaystyle 2772{ \ m }^{ 2 }$

  3. $\displaystyle 1772{\  m }^{ 2 }$

  4. $\displaystyle 2672{\  m }^{ 2 }$

Show answer
Correct Option: B
Explanation:
Given $r=14m,s=49m$
Surface area of a cone 
$=$ $\displaystyle \pi rs+\pi { r }^{ 2 }$

$\displaystyle =\dfrac {22}{7} \times 14\times 49+\dfrac {22}{7} \times 14\times 14$

$\displaystyle =2156+616$

$\displaystyle =2772{ m }^{ 2 }$

The total surface area of a conical jar is $\displaystyle 740{ ft }^{ 2 }$. If its slant height is two times the radius, then what is the base diameter of the colical jar? (use $\displaystyle \pi =3$).

maths solids surface area of a cone surface area of cone the surface area of a cone

  1. $18.12$ ft

  2. $18.10$ ft

  3. $18.24$ ft

  4. $18.31$ ft

Show answer
Correct Option: A
Explanation:

Formula:

Surface area of cone=$\pi rs+\pi r^2$
$s=2r$
where 
$r$ is the radius of the base of the cone.
$s$ is the slant height of the cone.
We know that surface area =$740\ ft^2$
$\therefore \pi rs+\pi r^2=740$
Substituting $s=2r$ and $\pi=3$ in the above equation we get,
$ 3 \times r \times 2r+3 r^2=740$
$\Rightarrow 3*2r^2+3r^2=740$
$\Rightarrow 6r^2+3r^2=740$
$\Rightarrow 9r^2=740$
$\Rightarrow r^2=\dfrac{740}{9}$
$\Rightarrow r^2=82.22$
$\Rightarrow r=\sqrt{82.22}$
$\Rightarrow r=9.06$
The diameter(d)=twice of radius
$\therefore d=2r$
$\therefore d=2 \times 9.06$
$\therefore d=18.12\ ft$

What is the total surface area of a cone if its diameter =${ 1}$ ft and slant height = $12$ ft. (use $\displaystyle \pi =3.14$).

maths solids surface area of a cone surface area of cone the surface area of a cone

  1. $\displaystyle 11.625{ ft }^{ 2 }$

  2. $\displaystyle 18.625{ ft }^{ 2 }$

  3. $\displaystyle 19.625{ ft }^{ 2 }$

  4. $\displaystyle 19.625ft$

Show answer
Correct Option: C
Explanation:

$\displaystyle Diameter=\frac { radius }{ 2 } $
$\displaystyle radius=\dfrac{1}{2}ft$
Surface area of a cone $\displaystyle =\pi rs+\pi { r }^{ 2 }$
$\displaystyle 3.14\times { 1 }/{ 2 }\times 12+3.14\times { 1 }/{ 2 }\times { 1 }/{ 2 }$
$\displaystyle 18.84+0.785$
$\displaystyle 19.625{\  ft }^{ 2 }$