Tag: maths

Questions Related to maths

The base of a right pyramid is an equilateral triangle of perimeter $8$ dm and the height of the pyramid is $30$$\sqrt{3}$ cm. The volume of the pyramid is

maths perimeter, area and volume volume of prism and pyramid surface areas and volumes of solids recall the surface areas and volumes of different solid shapes

  1. $1600$ cm$^{3}$

  2. $16000$ cm$^3$

  3. $\displaystyle \frac{16000}{3} cm^3$

  4. $\displaystyle \frac{5}{4} cm^3$

Show answer
Correct Option: B
Explanation:
Base of pyramid is an equilateral triangle of parameter
$8dm=80cm$
Let the side of the equilateral triangle be $'a'cm$
$\therefore $Parameter of equilateral triangle$=3a$
$\Rightarrow 3a=80\Rightarrow a=\cfrac { 80 }{ 3 } cm$
Height of pyramid$=30\sqrt { 3 } cm$
Volume of pyramid=Area of base $\times $ height
$=\cfrac { \sqrt { 3 }  }{ 4 } { a }^{ 2 }\times 30\sqrt { 3 } $

$=\cfrac { \sqrt { 3 }  }{ 4 } \times \cfrac { 80 }{ 3 } \times \cfrac { 80 }{ 3 } \times 30\sqrt { 3 } $

$=\cfrac { 3 }{ 4 } \times \cfrac { 80 }{ 3 } \times 80 \times 10 $

$=\cfrac { 1 }{ 4 } \times 80 \times 80 \times 10 $

$=20 \times 80 \times 10 $

$=16000cm^3$

If the areas of the adjacent faces of a rectangular block are in the ratio $2:3:4$ and its volume is $9000{cm}^{3}$, then the length of the shortest edge is

maths perimeter, area and volume volume of prism and pyramid surface areas and volumes of solids recall the surface areas and volumes of different solid shapes

  1. $30cm$

  2. $20cm$

  3. $15cm$

  4. $10cm$

Show answer
Correct Option: C
Explanation:

Let the edge of the cuboid be $a\,cm,\,b\,cm$ and $c\,cm$.

And, $a<b<c$
The areas of the three adjacent faces are in ratio $2:3:4$
So,
$ab:ca:bc=2:3:4$ and its volume is $9000\,cm^3$
We have to find the shortest edge of the cuboid
Since,
$\dfrac{ab}{bc}=\dfrac{2}{4}$

$\dfrac{a}{c}=\dfrac{1}{2}$

$\therefore$  $c=2a$
Similarly,
$\dfrac{ca}{bc}=\dfrac{3}{4}$

$\dfrac{a}{b}=\dfrac{3}{4}$

$\therefore$  $b=\dfrac{4a}{3}$

Volume of cuboid,
$V=abc$
$\Rightarrow$  $9000=a\left(\dfrac{4a}{3}\right)(2a)$

$\Rightarrow$  $27000=8a^3$

$\Rightarrow$  $a^3=\dfrac{27\times 1000}{8}$

$\Rightarrow$  $a=\dfrac{3\times 10}{2}$

$\therefore$  $a=15\,cm$
Now, $b=\dfrac{4a}{3}=\dfrac{4\times 15}{3}=20$
$c=2a=2\times 15=30\,cm$
$\therefore$  The length of the shortest edge is $15\,cm.$

A right pyramid is on a regular hexagonal base. Each side of the base is 10 m. Its height is 60 m.The volume of the pyramid is

maths perimeter, area and volume volume of prism and pyramid surface areas and volumes of solids recall the surface areas and volumes of different solid shapes

  1. 5196 $m^3$

  2. 5200 $m^3$

  3. 5210 $m^3$

  4. 51220$m^3$

Show answer
Correct Option: A
Explanation:

Volume of pyramid $= \displaystyle \frac{1}{3}$ [Area of hexagonal base of side 10 m $\times$ Height]
$= \displaystyle \frac{1}{3} \left [ \frac{3 \sqrt 3}{4} (10)^2 \times 60 \right ] = 5196 m^3$.

A right pyramid on a regular hexagonal base is of height $60$ m. Each side of the base is $10$ m. The volume of the pyramid is

maths perimeter, area and volume volume of prism and pyramid surface areas and volumes of solids recall the surface areas and volumes of different solid shapes

  1. $\displaystyle 4500\ \text{m}^{3}$

  2. $\displaystyle 5000\ \text{m}^{3}$

  3. $\displaystyle 5196\ \text{m}^{3}$

  4. $\displaystyle 6196\ \text{m}^{3}$

Show answer
Correct Option: C
Explanation:

Volume of the pyramid
$\displaystyle =\frac{1}{3}\times $ base area $\displaystyle \times $height
$\displaystyle =\frac{1}{3}\times \left ( \frac{3}{2}\sqrt{2}\times 10^{2} \right )\times 60m^{2}$,
since $\displaystyle \sqrt{3}=1.732$
=$\displaystyle =5196m^{3}$

A regular square pyramid is $3$ m height and the perimeter of its base is $16$ m. Find the volume of the pyramid.

maths perimeter, area and volume volume of prism and pyramid surface areas and volumes of solids recall the surface areas and volumes of different solid shapes

  1. 1212 $cu. m$

  2. 1414 $cu. m$

  3. 1616 $cu. m$

  4. 1818 $cu. m$

Show answer
Correct Option: C
Explanation:

Given, height of regular square pyramid is $3$ m and the perimeter of its base is $16$ m

Let the base side of pyramid is $l$ m
Then perimeter of base $=4a=16$
So, $ a=4$
Then volume of pyramid $=$ $\dfrac{1}{3}l^{2}h=\dfrac{1}{3}\times (4)^{2}\times 3=16 $ $cu. m$

The altitude of the frustum of a regular rectangular pyramid is $5\ m$ the volume is $140\ cu.\ m.$ and the upper base is $3\ m$ by $4\ m$. What are the dimensions of the lower base in $m$?

maths perimeter, area and volume volume of prism and pyramid surface areas and volumes of solids recall the surface areas and volumes of different solid shapes

  1. $9\times10$

  2. $6\times8$

  3. $4.5\times6$

  4. $7.5\times10$

Show answer
Correct Option: B
Explanation:

Given : Height of pyramid$(h)=5\ m$, Volume$(V)=140\ cu. m$

Edge of upper base are $length(L)=3\ m, breadth(B)=4\ m$
And let $l,h$ be the length and height of lower base

$\therefore$ Area of upper base$(A _{1})=L\times B=3\times 4 = 12\ m^2$
And Area of lower base$A _{2}=l\times b$
Volume of frustum$(V)=\dfrac{h}{3}(A _{1} + A _2 + \sqrt{A _1 \times A _2})$
$\implies$$140=\dfrac{5}{3}(12 + A _2 + \sqrt{12 \times A _2})$
$\implies 84=A _2 + 12 + \sqrt{12}\ \sqrt{A _2}$
$\implies A _2+\sqrt{12}\ \sqrt{A _2}-72=0$
$\implies \left(\sqrt{A _2} + \dfrac{\sqrt{12}}{2}\right)^2 - \dfrac{12}{4} - 72=0$
$\implies \left(\sqrt{A _2} + \dfrac{\sqrt{12}}{2}\right)^2 =75$
$\implies \left(\sqrt{A _2} + \dfrac{\sqrt{12}}{2}\right) =\sqrt{75}$
$\implies \sqrt{A _2}=6.928$
$\therefore A _2 = 47.9971=48\ m^2=6\times 8$ 
Hence, dimensions of lower base is $6 \times 8$.

The length of the base of a square pyramid is $2\ cm$ and the height is $6\ cm$. Calculate the volume.

maths perimeter, area and volume volume of prism and pyramid surface areas and volumes of solids recall the surface areas and volumes of different solid shapes

  1. $8\ cm^3$

  2. $6\ cm^3$

  3. $4\ cm^3$

  4. $2\ cm^3$

Show answer
Correct Option: A
Explanation:

Volume of square pyramid  $=\dfrac { 1 }{ 3 } \times { a }^{ 2 }\times h=\dfrac { 1 }{ 3 } \times 2\times 2\times 6=8{ cm }^{ 3 }$

The base of a right pyramid is an equilateral triangle of perimeter 8 cm and the height of the pyramid is $30\sqrt 3$ cm. The volume of the pyramid is

maths perimeter, area and volume volume of prism and pyramid surface areas and volumes of solids recall the surface areas and volumes of different solid shapes

  1. $160 cm^3$

  2. $1600 cm^3$

  3. $\dfrac {160}{3} cm^3$

  4. $\dfrac {5}{4} cm^3$

Show answer
Correct Option: A
Explanation:

Volume of right pyramid $=$ $\dfrac { 1 }{ 3 } \times area\quad of\quad base\times height\quad of\quad pyramid$

Now, base is equilateral $\triangle $, therefore,
area $=\dfrac { \sqrt { 3 }  }{ 4 } \times { \left( side \right)  }^{ 2 }$
Perimeter of triangle $=8cm$
$\therefore \quad \quad 3a=8\Rightarrow a=\dfrac { 8 }{ 3 } cm$
$\therefore \quad \quad area=\dfrac { \sqrt { 3 }  }{ 4 } \times \dfrac { 8 }{ 3 } \times \dfrac { 8 }{ 3 } =\dfrac { 16\sqrt { 3 }  }{ 9 } { cm }^{ 2 }$
Now,  Volume $=\dfrac { 1 }{ 3 } \times \dfrac { 16\sqrt { 3 }  }{ 9 } \times 30\sqrt { 3 } $
                        $=\dfrac { 160\times 3 }{ 9 } =\dfrac { 160 }{ 3 } { cm }^{ 3 }$

A right pyramid is on a regular hexagonal base. Each side of the base is $10$ m. Its height is $60$ m. The volume of the pyramid is

maths perimeter, area and volume volume of prism and pyramid surface areas and volumes of solids recall the surface areas and volumes of different solid shapes

  1. $5196 m^3$

  2. $5200 m^3$

  3. $5210 m^3$

  4. $5220 m^3$

Show answer
Correct Option: A
Explanation:
Volume of a hexagonal Pyramid $=$ $\displaystyle \frac{\sqrt{3}}{2}a^2h$
where $a$ $=$ side of the base. and $h$ $=$ height of the pyramid

$\therefore $Volume of a pyramid $=$ $\displaystyle \frac{\sqrt{3}}{2}{(10^2)}\times {60}$ $=$ $5196$ $m^3$

If a regular square pyramid has  a base of side 8 cm and height  of 30 cm, then its volume is

maths perimeter, area and volume volume of prism and pyramid surface areas and volumes of solids recall the surface areas and volumes of different solid shapes

  1. 120 c.c.

  2. 240 c.c.

  3. 640 c.c.

  4. 900 c.c.

Show answer
Correct Option: C
Explanation:

Side of square base $=$ 8 cm.
Height of pyramid $=$ 30 cm
$\therefore$ Volume of square pyramid
$= \frac{1}{3}\times $area  of  base $\times$ height
$= \frac{1}{3}\times 82\times 30 = 640$ c.c.