Tag: solids

Questions Related to solids

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

  2. $\pi rl$

  3. $\pi rl^2$

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

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Correct Option: B
Explanation:

Consider the given slant height l and radius$ =r$ of the cone,

 Curved surface area=(Arc length of sector/Circumference of circle)×Area of circle Curved surface area

$=\dfrac{2\pi r}{2\pi l}\times \pi {{l}^{2}}=\pi rl$ 




If a sphere has the same curved surface area as total surface area of cone of vertical height 40 cm and radius 30 cm then the radius of the sphere is

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

  1. $ \displaystyle 10\sqrt{6} $ cm

  2. $ \displaystyle 10\sqrt{3} $ cm

  3. $ \displaystyle 10\sqrt{2} $ cm

  4. 12 cm

Show answer
Correct Option: B
Explanation:

Given the vertical height of cone is 40 cm and radius is 30 cm 

Then  surface area of cone=$\pi rh=\pi \times 30\times 40=1200 \pi cm^{2}$
The surface area of cone =covered surface area of sphere
$\therefore \pi R^{2}=1200 \pi \Rightarrow R^{2}=300\Rightarrow R=10\sqrt{3}cm$
Then the radius of sphere =$10\sqrt{3}cm$

The cost of canvas required for a conical tent of height 8 m and diameter of base 12 m at the rate of Rs 3.50 per $\displaystyle m^{2}$ is

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

  1. RS 620

  2. Rs 600

  3. Rs 640

  4. Rs 660

Show answer
Correct Option: D
Explanation:

Given that conical tent of canvas height is 8 cm and diameter of base is 12 m

Then radius of base =$\frac{12}{2}=6 m$
Then area of conical tent =$\pi r(\sqrt{r^{2}+h^{2}})= \frac{22}{7}\times 6\times \sqrt{(10)^{2}+(6)^{2}}\Rightarrow 60\times \frac{22}{7}cm^{2}$
If cost of canvas is 3.50 per sq cm 
Then cost of canvas=$\times 60\times \frac{22}{7}\times 3.5= Rs 660 $

The volume of the cone whose vertical height is 8 m and the area of base $ \displaystyle 156  m^{2}     $ is

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

  1. 416 $ \displaystyle m^{2} $

  2. 415 $ \displaystyle m^{2} $

  3. 312 $ \displaystyle m^{2} $

  4. 468 $ \displaystyle m^{2} $

Show answer
Correct Option: A
Explanation:

Given the vertical height of cone is 8 m and area of base is 156 sq m

Let radius of cone is r m 
Then base area =$\pi r^{2}=156\Rightarrow r^{2}=156 \pi  m$
And the volume of cone=$\frac{1}{3}\pi r^{2}h=\frac{1}{3}\times 156\pi \times 8=416 m^{3}$

A conical tent of radius of 12 m and height 16 m is to be made, then the cost of canvas required at the rate Rs 10 per $ \displaystyle   m^{2}   $ is

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

  1. RS 7445

  2. Rs 7543

  3. Rs 7550

  4. Rs 7500

Show answer
Correct Option: B
Explanation:

Given the radius a of height of tent is 12 m and 16 m respt 

Then slant height of tent s=$\sqrt{r^{2}+h^{2}}=\sqrt{(12)^{2}+(16)^{2}}=\sqrt{144+256}=\sqrt{400}=20 m$
Then covered surface area of tent=$\pi rs=3.143\times 12\times 20=754.30 m^{2}$
If cost of canvas is Rs 10 per sq m
Then total cost =$754.30\times10=7543 RS

The curved surface area of a right circular cone of height 84 cm and diameter 70 cm is

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

  1. 10010 $ \displaystyle cm^{2} $

  2. 100000 $ \displaystyle cm^{2} $

  3. 10020 $ \displaystyle cm^{2} $

  4. 11000 $ \displaystyle cm^{2} $

Show answer
Correct Option: A
Explanation:

Given 84 cm is the height of right circular cone and diameter is 70 cm 

Then radius of  right circular cone=$\frac{70}{2}=35 cm$
Then curved surface area of a right circular cone =$\pi r(\sqrt{r^{2}+h^{2}})=\frac{22}{7}\times 35\left ( \sqrt{(35)^{2}+(84)^{2}} \right )=110\left ( \sqrt{7056+1225} \right )=110\times 91=10010$ sq cm

The slant height of a right circular cone is 10 m and its height is 8 cm then the area of its curved surface is

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

  1. 80 $ \displaystyle \pi m^{2} $

  2. 60 $ \displaystyle \pi m^{2} $

  3. 65 $ \displaystyle \pi m^{2} $

  4. 70 $ \displaystyle \pi m^{2} $

Show answer
Correct Option: B
Explanation:

Given that slant height of a right circular cone is 10 cm and its height is 8 cm 

Then radius of cone $R^{2}=(10)^{2}-(8)^{2}=100-64=36\Rightarrow R=6 cm$
Then curved surface area of right circular cone =$\pi rl=\pi \times 6\times 10==60\pi cm^{2}$

The total area of sheet required to make an open cone of height 24 cm and radius 7 cm is

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

  1. 470 $ \displaystyle cm^{2} $

  2. 450 $ \displaystyle cm^{2} $

  3. 425 $ \displaystyle cm^{2} $

  4. 550 $ \displaystyle cm^{2} $

Show answer
Correct Option: D
Explanation:

Given height of cone is 24 cm and radius is 7 cm 

Then sheet required = surface area of cone =$\pi r(\sqrt{r^{2}+h^{2}})=2\times \frac{22}{7}\times 7(\sqrt{(24)^{2}+(7)^{2}})=2\times 22\times 25=550 cm^{2}$

If the diameter of a right cone is 6 cm and its vertical height is 4 cm then its curved surface area is

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

  1. 47.1 $ \displaystyle cm^{2} $

  2. 48 $ \displaystyle cm^{2} $

  3. 49 $ \displaystyle cm^{2} $

  4. 50 $ \displaystyle cm^{2} $

Show answer
Correct Option: A
Explanation:

Given the vertical height is 4 cm and diameter is 6 cm

Then radius =$\frac{6}{2}=3$
And slant height =$l=\sqrt{r^{2}+h^{2}}=\sqrt{(3)^{2}+(4)^{2}}=\sqrt{9+16}=\sqrt{25}=5$ cm
Then curved surface area of cone =$\pi rL=\pi \times 3\times 5=3.14\times 5\times 3=47.1 cm^{2}$

If the area of the base of a right circular cone is 51 $ \displaystyle m^{2}     $and volume is 68 $ \displaystyle m^{3}     $ then its vertical height is

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

  1. 3.5 m

  2. 4 m

  3. 4.5 m

  4. 5 m

Show answer
Correct Option: B
Explanation:

Given the area of  base of right circular cone is 51 sq m and volume is 68 cu m

Then area of base =$\pi r^{2}=51$
$\Rightarrow r^{2}=\frac{51}{\pi }$
$\therefore volume =\frac{1}{3}\pi r^{2}h=68$
$\Rightarrow\frac{1}{3} \pi \left ( \frac{51}{\pi } \right )h= 68$
$\Rightarrow h=\frac{68}{17}$
$\Rightarrow h=4$ m