Tag: solids

Questions Related to solids

The radii of two right circular cone are int eh ratio of 4 : 5 and their slant heights are in the ratio 2:3 Then the ratio of their curved surfaces is

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

  1. 6 : 17

  2. 8 : 15

  3. 1 : 1

  4. none of these

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Correct Option: B
Explanation:

Let the radii of two right circular cone is 4x and 5x and siant height is 2y and 3y

Then curved surface area of first  right circular cone=$\pi \times 4x\times 2y=8\pi xy$ sq cm
And curved surface area of second right circular cone=$\pi \times 5x\times 3y=15\pi xy$ sq cm
So ratio of both right circular cone=$8\pi xy:15\pi xy::8:15$

The circumference of the base of a $24$ m high conical tent is $44$ m Calculate the length of canvas used in making the tent if the width of the canvas is $2$ m

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

  1. $257$ m

  2. $275$ m

  3. $752$ m

  4. $285$ m

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Correct Option: B
Explanation:

Let the radius of the base be r meters Then
$2$$\displaystyle \pi r$$=44$ m
or $\displaystyle r=\frac{44}{2\pi }=\frac{44\times 7}{2\times 22}=7 m$
Let the slant height be l meters Then
$\displaystyle l=\sqrt{h^{2}+r^{2}}=\sqrt{7^{2}+24^{2}}=\sqrt{625}=25m$
Now, Surface area = $\displaystyle \pi rl=\frac{22}{7}\times 7\times 25=550 m^{2}$
$\displaystyle \therefore $ Area of canvas required $= 550$ m$\displaystyle ^{2}$
 Given that width of the canvas is $2$ m therefore
Length of canvas required = $\displaystyle \frac{550}{2}=275m$

The slant height of a cone is $ 40$  m, the radius of the base is  $12$ m. Find the curved surface area of a cone.

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

  1. $\displaystyle 1507.2{ \ mm }^{ 2 }$

  2. $\displaystyle 1507.2\ m$

  3. $\displaystyle 1507.2{\  m }^{ 2 }$

  4. $\displaystyle 1407.2{\  mm }^{ 2 }$

Show answer
Correct Option: C
Explanation:

Using the forrmula for Curved surface area of the cone$= \pi rs$

where, $\pi=3.14$
$radius(r)=12\ m$
$slant\ height(s)=40\ m$
$\therefore$ Curved surface area$=3.14 \times 12 \times 40=1507.2m^2$

The formula used for surface area of cone is  ($s$ denotes slant height.)

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

  1. $\displaystyle \pi r\left( r+s \right) $ sq.units

  2. $\displaystyle \pi r\left( 2r+s \right) $ sq.units

  3. $\displaystyle 2\pi r\left( r+s \right) $ sq.units

  4. $\displaystyle \pi r{ \left( r+s \right) }^{ 2 }$ sq.units

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Correct Option: A
Explanation:

Surface area of a cone $=$ $\displaystyle \pi r\left( r+s \right) $ sq.units
Since, surface area of cone $=$  Area of sector $+$ Area of circle.

The curved surface area of the cone is $\pi r l$ whereas, total surface area of the cone is

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

  1. $\pi r(r+l)$

  2. $\pi rl(r+l)$

  3. $2\pi rl(r-l)$

  4. $2\pi r^2l(r+l)$

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Correct Option: A
Explanation:

Surface area of cone $=$ CSA of cone + area of circular base

                                    $=$ $\pi rl+\pi { r }^{ 2 }\Rightarrow \pi r(l+r)$

If the circumference at the base of a right circular cone and the slant height are $120\pi$ $cm$ and $10cm$ respectively, then the curved surface area of the cone is equal to

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

  1. $1200\pi$ ${cm}^{2}$

  2. $600\pi$ ${cm}^{2}$

  3. $300\pi$ ${cm}^{2}$

  4. $600$ ${cm}^{2}$

Show answer
Correct Option: B
Explanation:

We have,

circumference $=120\pi\ cm$, slant height$(l)=10\ cm$
$\Rightarrow 2\pi r=120\pi$
$\Rightarrow 2r=120$
$\Rightarrow r=60\ cm$

We know that the curved surface area of the cone
$=\pi r l$

So,
$=\pi \times 60 \times 10$

$=600\pi\ cm^2$

Hence, this is the answer

Two right circular cones have equal radii. If their slant heights are in the ratio $4:3$, then their respective curved surface areas are in the ratio

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

  1. $16:9$

  2. $2:3$

  3. $4:3$

  4. $3:4$

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Correct Option: C
Explanation:

We have,

The radius of the two cone are equal

The ratio of the slant height $l _1:l _2=4:3$

We know that the curved surface area of the cone
$=\pi r l$

So, the required ratio

$=\dfrac{\pi r l _1}{\pi r l _2}$

$=\dfrac{l _1 }{l _2 }$

$=\dfrac{4}{3}=4:3$

Hence, this is the answer.

The curved surface area of a right circular cone of height $15$ cm and base diameter $16$ cm is __________.

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

  1. $60\pi \ \text{cm}^2$

  2. $68\pi \ \text{ cm}^2$

  3. $120\pi \ \text{cm}^2$

  4. $136\pi \ \text{cm}^2$

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Correct Option: D
Explanation:

The formula of curved surface area of a right circular cone $ =\pi \times r\times l $

$ \Rightarrow l=\sqrt { { (h }^{ 2 } } +{ r }^{ 2 }) $
$ \Rightarrow l=\sqrt { 8^{ 2 } } +{ 15 }^{ 2 }) $
$ \Rightarrow l=17 $ cm
Now substitute the value in above equation. we have
Curved surface area $ =\pi \times 8\times 17 $
$ \Rightarrow 136\pi \ { \text{cm} }^{ 2 } $
So, option D is the correct.

The area of the base of a cone is $616\, cm^2$ and its height is 48 cm. The total surface area of cone is 

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

  1. $2816\, cm^2$

  2. $2861\, cm^2$

  3. $2618\, cm^2$

  4. $2681 \, cm^2$

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Correct Option: A
Explanation:
Given the area of the base of  a cone is $616\ cm^2$.
If $r$ be the radius of the base of the cone then 
$\pi r^2=616$
or, $\dfrac{22}{7}\times r^2=616$
or, $r=14$.
Also, given height $(h)=48\ cm$.
Then slant height $(l)=\sqrt{48^2+14^2}=2\sqrt{625}=50\ cm$.
$\therefore$ total surface area of the cone $=\pi r(r+l)=\dfrac{22}{7}\times 14\times 64=2816\ cm^2$.

The base radii of a cone and a cylinder are equal. If their curved surface areas are also equal, then the ratio of the slant height of the cone to the height of the cylinder is

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

  1. $2 : 1$

  2. $1 : 2$

  3. $1 : 3$

  4. $3 : 1$

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Correct Option: A
Explanation:
Let the radius of the cone and cylinder be $r$.
The base radii of cone and cylinder are equal.
Given curved surface areas are equal,
$\therefore (\pi)rl = 2(\pi)rh$
$\therefore \dfrac {l}{h}=2$
Hence, option A is correct.