Tag: solids

Questions Related to solids

An ice cream cone has radius of $21$ m and height of $20$ m. 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. $\displaystyle 3297{\  m }^{ 2 }$

  2. $\displaystyle 9734{ \ cm }^{ 2 }$

  3. $\displaystyle 3297\ mm$

  4. $\displaystyle 3297\ m$

Show answer
Correct Option: A
Explanation:

Find the value of slant height(s) of a ice-cream cone, using Pythagoras theorem, since the cross section is a right triangle.
$\displaystyle { s }^{ 2 }={ h }^{ 2 }+{ r }^{ 2 }$
$\displaystyle { s }^{ 2 }={ 20 }^{ 2 }+{ 21 }^{ 2 }$
$\displaystyle { s }^{ 2 }=400+441$
$\displaystyle s=\sqrt { 841 } $
$\displaystyle s=29$ m
So, surface area of a cone $\displaystyle =\pi rs+\pi { r }^{ 2 }$
$\displaystyle =3.14\times 21\times 29+3.14\times 21\times 21$
$\displaystyle =1912.26+1384.74$
$\displaystyle =3297{ m }^{ 2 }$

What is the total surface area of a cone, if its radius = 5 cm and height = $\displaystyle\sqrt2 $cm?

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

  1. $\displaystyle 159.983{ \ mm }^{ 2 }$

  2. $\displaystyle 159.983{\  cm }^{ 2 }$

  3. $\displaystyle 159.983\ cm$

  4. $\displaystyle 159.983{ \ m }^{ 2 }$

Show answer
Correct Option: B
Explanation:

The value of slant height(s) of a cone, using Pythagoras theorem, since the cross section is a right triangle is

$s^2=h^2+r^2$
where 
h= height of cone
r=radius of the base of the cone
we know that
$h=\sqrt2\ cm$
$r=5\ cm$
$\therefore s^2=\sqrt2^2+5^2=2+25=27$
$\Rightarrow s^2=27$
$\Rightarrow s=\sqrt27$
$\Rightarrow s=5.19\ cm$

Total Surface area of cone$=\pi rs+\pi r^2$
$\pi=3.14$
$r=5\ cm$
$s=5.19\ cm$
$\therefore \pi rs+\pi r^2=3.14\times 5\times5.19+3.14\times 5\times5$
$=81.483+78.5$
$=159.983\ cm^2$

The total surface area of a conical tent is $ 920$ square meter and its radius $14$ m. Find the slant height. (Round off your answer to the nearest whole number).

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

  1. $7$ m

  2. $6$ m

  3. $5$ m

  4. $6.5$ m

Show answer
Correct Option: A
Explanation:

Formula:

Surface area of cone$=\pi rs+\pi r^2$
where r is the radius of the base of the cone and s is the slant height.
We know that surface area of cone$=940\ m^2$
$r =14\ m$
$\pi=3.14$
Substituting the values in the formula we get
$\Rightarrow 940=3.14 \times 1\times 4s+3.14\times 14^2$
$\Rightarrow 940=43.96\times s+3.14\times 196$
$\Rightarrow 940=43.96\times s+615.44$
$\Rightarrow 940-615.44=43.96\times s$
$\Rightarrow 324.56=43.96\times s$
$\Rightarrow s=\dfrac{324.56}{43.96}$
$\Rightarrow s=7.38\approx 7\ m$

A closed cone tank radius $7$ cm and height $10$ cm is made from a sheet of aluminium. How much sheet is required?

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

  1. $144cm^2$ 

  2. $\displaystyle 22\sqrt { 149 } cm^2$

  3. $\displaystyle (22\sqrt { 149 } +144) cm^2$

  4. None of the above

Show answer
Correct Option: C
Explanation:

Total surface area of cone  = Area of base + Area of curved surface
$\displaystyle =\quad \pi { r }^{ 2 }+\pi rs$

$\displaystyle =\frac { 22 }{ 7 } \times 7\times 7+\frac { 22 }{ 7 } \times 7\left( \sqrt { 149 }  \right) $
$\displaystyle s=\sqrt { { h }^{ 2 }+{ r }^{ 2 } } $
$\displaystyle =\sqrt { 149+100 } $
$\displaystyle =144+22\sqrt { 149 } $

Find  the radius of the base of a right circular cone which has a lateral surface area of $6\pi$ and a slant height of $6$ ( in standard units )

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

  1. $0.50$

  2. $0.75$

  3. $1.00$

  4. $1.25$

Show answer
Correct Option: C
Explanation:

Given, Lateral surface area $=6 \pi$ and slant height $=6$

Let the radius of base be $r$ and slant height of cone be $l = 6$.
Lateral surface area is equal to $\pi rl = \pi \times r \times 6 = 6\pi$
$\Rightarrow 6 \pi= \pi \times r \times 6$
$\Rightarrow r=1$

Find the radius of the base of a cone having a slant height of $8$ and a lateral area of $48\pi$. 

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

  1. $3$

  2. $6$

  3. $12$

  4. $16$

  5. $2$

Show answer
Correct Option: B
Explanation:
Given, slant height $l = 8$
Lateral area of cone is $\pi rl = 48 \pi$
$\Rightarrow rl = 48$
$\Rightarrow 8r=48$
$\Rightarrow r = \dfrac {48}{8} = 6$

Find the slant height and vertical height of a Cone with radius $5.6$ cm and curved surface area $158.4$ cm$^2$.

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

  1. $8.07$

  2. $7.05$

  3. $8$

  4. None of the above

Show answer
Correct Option: B
Explanation:

Radius $=5.6$ cm, vertical height $= h$, slant height $=$ $l$
Curved Surface Area of cone $=\pi r l = 158.4 cm^2$
$\Rightarrow \dfrac{22}{7} \times 5.6 \times l = 158.4$
$\Rightarrow l = \dfrac{158.4 \times 7}{22 \times 5.6} = \dfrac{18}{2} = 9$ cm
We know $l^2 = r^2 + h^2$
Thus $h^2 = l^2 - r^2 $

$= 9^2 - (5.6)^2$
$= 81 - 31.36$
$= 49.64$
$h = \sqrt{49.64}$
$h = 7.05 $ cm (approx.)

Find the area of canvas required for a conical tent whose height is $3.5$ m and the radius of the base is $12$ m.

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

  1. $271.42$

  2. $471.42$

  3. $371.42$

  4. $571.42$

Show answer
Correct Option: A
Explanation:

$l^2=(3.5)^2+12^2=156.25$
$l=12.5$
Curved surface area$=\pi rl$
$=471.42m^2$

The curved surface area of right circular cone with height $24$ m and radius $7$ m is

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

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

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

  3. $607\ \text{m}^{ 2 }$

  4. $650\ \text{m}^{ 2 }$

Show answer
Correct Option: B
Explanation:
Height of cone$=24m$
Radius of cone$=7m$
Slant height of cone$=\sqrt { { 24 }^{ 2 }+{ 7 }^{ 2 } } =\sqrt { 576+49 } =\sqrt { 625 } =25m$
CSA of cone$=\pi rl=\cfrac { 22 }{ 7 } \times 7\times 25=550 m^2$

A cone and a cylinder have the same base area. They also have the same curved surface area. If the height of the cylinder is $3$ m, then the slant height of the cone (in m) is

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

  1. $3$

  2. $4$

  3. $6$

  4. $7$

Show answer
Correct Option: C
Explanation:

Given the radius of cylinder and cone are the same because there base areas are same.
Curved surface of cylinder $=$ curved surface area of the cone
$\therefore 2 \pi r h = \pi r l$
$\therefore l = 2h $
Given, height of cylinder $= 3$ cm
Therefore, slant height of cone $= 6$ cm