Tag: solids

Questions Related to solids

A circus tent is in the form of a cone over a cylinder. The diameter of the base is $9$ m, the height of cylindrical part is $4.8$ m and the total height of the tent is $10.8$ m. The canvas required for the tent is ..........

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

  1. $24.184$ sq.m

  2. $2418.4$ sq.m

  3. $241.84$ sq.m

  4. None of these

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Correct Option: C
Explanation:

The amount of canvas used to make the tent $ = $ Curved surface area of cylindrical part $ + $ Curved surface area of the conical part.
Curved Surface Area of a Cylinder of Radius "$R$" and height "$h$" $ = 2\pi Rh$
Radius of the cylindrical part $ = \dfrac {Diameter}{2} = \dfrac {9}{2} $ m 
Curved surface area of a cone $= \pi rl$, where $r$ is the radius of the cone and $l$ is the slant height.

Radius of the conical part $ = \dfrac {Diameter}{2} = \dfrac {9}{2}  m $

Height of the conical part $ = 10.8 - 4.8 = 6 $ m 

For a cone, $  l= \sqrt { { h }^{ 2 }+  {r}^{ 2 } } $, where $h$ is the height. 

Hence, $l = \sqrt { { 6 }^{ 2 }+  {4.5}^{ 2 } } $

$\therefore  l = 7.5  $ m 

Hence, area of canvas $ = 2 \times \pi \times 4.5 \times 4.8 + \pi \times 4.5 \times 7.5$

$ = 76.95 \pi  $

$= 241.84  {m}^{2} $

The radius of a cone is $3\ cm$ and vertical height is $4\ cm$. Find the area of the curved surface.

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

  1. $62.85\ {cm}^{2}$

  2. $66.05\ {cm}^{2}$

  3. $52.25\ {cm}^{2}$

  4. $47.14\ {cm}^{2}$

Show answer
Correct Option: D
Explanation:

Curved surface area of a cone $= \pi rl$

For a cone, $l = \sqrt { {h }^{ 2 }+  {r}^{ 2 } } $

Hence, $l = \sqrt { { 3 }^{2 }+  {4}^{ 2 } } $

$ l = 5  cm $
So, CSA of this cone, $ = \cfrac {22}{7} \times 3 \times 5 = 47.14  {cm}^{2} $

The radius of the base of a conical tent is $7\space m$. The tent is $24\space m$ high. Find the cost of the canvas required to make the tent, if one square meter of canvas costs $Rs. 180$ (Take $\pi = 3.14$)

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

  1. $Rs. 99000$

  2. $Rs. 98000$

  3. $Rs. 95000$

  4. $Rs. 97000$

Show answer
Correct Option: A
Explanation:

Radius $= 7m$,  height $= 24m$

CSA of tent = $\pi rl$
Now, $l=\sqrt { { r }^{ 2 }+{ h }^{ 2 } } =\sqrt { { 7 }^{ 2 }+{ 24 }^{ 2 } } =\sqrt { 49+576 } =\sqrt { 625 } =25cm$
Now, CSA of tent = $\dfrac { 22 }{ 7 } \times 7\times 25=550m$
Therefore, cost of canvas = Rs.$(550$$\times $$180) = Rs.99000$

The radius and height of a cone are in the ratio $4:3$. The area of the base is $154\space cm^2$. Find the area of the curved surface.

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

  1. $192.5\space cm^2$

  2. $195\space cm^2$

  3. $190.5\space cm^2$

  4. $185.5\space cm^2$

Show answer
Correct Option: A
Explanation:

Let radius be $4x$ and height be $3x$.


$\therefore $ Area of base $=$ $\pi { r }^{ 2 }=154\Rightarrow \dfrac { 22 }{ 7 } \times { \left( 4x \right)  }^{ 2 }=154$


$\Rightarrow \quad 16{ x }^{ 2 }=\dfrac { 154\times 7 }{ 22 } \Rightarrow { x }^{ 2 }=\dfrac { 49 }{ 16 } \Rightarrow x=\dfrac { 7 }{ 4 } $

$\therefore $ Radius $=$ $4\times \dfrac { 7 }{ 4 } =7cm,\quad height=3\times \dfrac { 7 }{ 4 } =\dfrac { 21 }{ 4 } =5.25cm$

Now, CSA of cone $=$ $\pi rl\quad and\quad l=\sqrt { { r }^{ 2 }+{ h }^{ 2 } } =\sqrt { 49+{ \left( 5.25 \right)  }^{ 2 } } $

$\therefore $ CSA $=$ $\dfrac { 22 }{ 7 } \times 7\times 8.75=192.5{ cm }^{ 2 }$

The curved surface area of of a right cone is $\displaystyle 286m^{2}$ and its slant height is 13 m then area of the base is

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

  1. $\displaystyle 286m^{2}$

  2. $\displaystyle 308m^{2}$

  3. $\displaystyle 154m^{2}$

  4. $\displaystyle 187m^{2}$

Show answer
Correct Option: C
Explanation:

Curved surface area of a cone $= \pi rl$  where r is the radius of
the cone and l is the slant height.
Hence, CSA of this cone, $ = \frac {22}{7} \times r \times 13 = 286 $
$ =>r = 7  m $

Area of base $ = \pi {r}^{2} = \frac {22}{7} \times 7 \times 7 = 154  m^2 $

The total surface area of cone if its slant height is 9 m, and the radius of its base is 12 m is

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

  1. 525 $\displaystyle cm^{2}$

  2. 792 $\displaystyle cm^{2}$

  3. 684 $\displaystyle cm^{2}$

  4. 412 $\displaystyle cm^{2}$

Show answer
Correct Option: B
Explanation:

We know that the total surface area S of a right circular cylinder of radius r and slant height l is given by 
$\displaystyle S=\pi r^{2}+\pi rl=\pi r\left ( r+l \right )$
Here, $\displaystyle r=12m\, and\, l=9m$
$\displaystyle \therefore S=\left { \frac{22}{7}\times 12\times \left ( 12+9 \right ) \right }^{2}=792m^{2}$

The curved surface area of a cone of slant height l and radius r is given by

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

  1. $ \displaystyle \frac{1}{3}\pi / r^{2} $

  2. $ \displaystyle \pi rl $

  3. $ \displaystyle \pi rl^{2} $

  4. $ \displaystyle \frac{1}{3}\pi rl $

Show answer
Correct Option: B
Explanation:

A cone is a three-dimensional geometric shape consisting of all line segments joining a single point to every point of a two-dimensional figure.

Slant height of cone (l)=$\sqrt{r^{2}+h^{2}}$Them covered surface area of cone =$\pi rl$

A cylindrical rod of lenght h is meted and cast into a cone of base radius twice that of the cylinder What is the height of the cone?

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

  1. $\displaystyle \frac{3h}{4}$

  2. $\displaystyle \frac{4h}{4}$

  3. 2h

  4. $\displaystyle \frac{h}{2}$

Show answer
Correct Option: A
Explanation:

Let the radius of  cylindrical rod is r and  cylindrical rod height is h given

Then volume of  cylindrical rod=$\pi r^{2}h$
And volume of cone of base twice the radius of   cylindrical rod=$\frac{1}{3}\pi (2r)^{2}H=\frac{4}{3}\pi r^{2}H$
The cylindrical rod melted and make cone 
$\therefore \frac{4}{3}\pi r^{2}H=\pi r^{2}h\Rightarrow H=\frac{3h}{4}$

The canvas required to construct a cone of height $24$ m and base radius $7$ m is

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

  1. $500$ $ \displaystyle \ \text{m} ^{2} $

  2. $520$ $ \displaystyle \ \text{m} ^{2} $

  3. $550$ $ \displaystyle \ \text{m} ^{2} $

  4. none of these

Show answer
Correct Option: C
Explanation:

Given the height of cone 24 m and base radius is 24 m

Then slant height =$\sqrt{r^{2}+h^{2}}=\sqrt{(24)^{2}+(7)^{2}}=\sqrt{576+49}=\sqrt{625}=25$
Then canvas required to constrict con=the curved surface area of cone=$\frac{22}{7}\times 7\times 25=550 cm^{2}$

Find the surface area of a cone whose radius is $12$ m and the slant height is $23$ m.

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

  1. $\displaystyle 420\pi { \ mm }^{ 2 }$

  2. $\displaystyle 420\pi {\  cm }^{ 2 }$

  3. $\displaystyle 420\pi \ { m }^{ 2 }$

  4. $\displaystyle 420\pi \ m$

Show answer
Correct Option: C
Explanation:
Surface area of cone is $A=πr(r+l)$

Here, radius is $r=12$ m and slant height is $l=23$ cm.

Thus,

$A=πr(r+l)=π\times 12(12+23)=12π\times 35=420π$ m$^2$

Hence, the surface area of the cone is $420π$ m$^2$.