Tag: recall the surface areas and volumes of different solid shapes

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Multiple choice 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

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

  1. $30cm$
  2. $20cm$
  3. $15cm$
  4. $10cm$
Reveal answer Fill a bubble to check yourself
C Correct answer
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 $15\,cm.$

Multiple choice 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

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

  1. 5196 $m^3$
  2. 5200 $m^3$
  3. 5210 $m^3$
  4. 51220$m^3$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

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

Multiple choice 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

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

  1. $\displaystyle 4500\ \text{m}^{3}$
  2. $\displaystyle 5000\ \text{m}^{3}$
  3. $\displaystyle 5196\ \text{m}^{3}$
  4. $\displaystyle 6196\ \text{m}^{3}$
Reveal answer Fill a bubble to check yourself
C Correct answer
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 $\displaystyle \sqrt{3}=1.732$
=$\displaystyle =5196m^{3}$

Multiple choice 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

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

  1. <span class="MathJax_Preview"><span class="MathJax"><span class="math"><span class="mrow"><span class="mn">12<span class="MJX_Assistive_MathML">12 $cu. m$
  2. <span class="MathJax_Preview"><span class="MathJax"><span class="math"><span class="mrow"><span class="mn">14<span class="MJX_Assistive_MathML">14 $cu. m$
  3. <span class="MathJax_Preview"><span class="MathJax"><span class="math"><span class="mrow"><span class="mn">16<span class="MJX_Assistive_MathML">16 $cu. m$
  4. <span class="MathJax_Preview"><span class="MathJax"><span class="math"><span class="mrow"><span class="mn">18<span class="MJX_Assistive_MathML">18 $cu. m$
Reveal answer Fill a bubble to check yourself
C Correct answer
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 $ $cu. m$

Multiple choice 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

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 $m$?

  1. $9\times10$
  2. $6\times8$
  3. $4.5\times6$
  4. $7.5\times10$
Reveal answer Fill a bubble to check yourself
B Correct answer
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 $6 \times 8$.

Multiple choice 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

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

  1. $8\ cm^3$
  2. $6\ cm^3$
  3. $4\ cm^3$
  4. $2\ cm^3$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

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

Multiple choice 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

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

  1. $160 cm^3$
  2. $1600 cm^3$
  3. $\dfrac {160}{3} cm^3$
  4. $\dfrac {5}{4} cm^3$
Reveal answer Fill a bubble to check yourself
A Correct answer
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 } $
                        $=\dfrac { 160\times 3 }{ 9 } =\dfrac { 160 }{ 3 } { cm }^{ 3 }$

Multiple choice 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

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

  1. $5196 m^3$
  2. $5200 m^3$
  3. $5210 m^3$
  4. $5220 m^3$
Reveal answer Fill a bubble to check yourself
A Correct answer
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 $=$ $\displaystyle \frac{\sqrt{3}}{2}{(10^2)}\times {60}$ $=$ $5196$ $m^3$
Multiple choice 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

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

  1. 120 c.c.

  2. 240 c.c.

  3. 640 c.c.

  4. 900 c.c.
Reveal answer Fill a bubble to check yourself
C Correct answer
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.

Multiple choice 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

If the volume of a prism is $1920$ $\sqrt{3} cm^3$ and the side of the equilateral base is $16$ $cm$, then the height (in cm) of the prism is?

  1. $19$
  2. $20$
  3. $30$
  4. $40$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation
Volume of prism $=$ Area of equilateral triangle $\times $ height
Now, Area of triangle $= \dfrac{\sqrt{3}}{4}a^2$
= $\dfrac{\sqrt{3}}{4} 16^2$ = $64 \sqrt{3}$
$1920 \sqrt{3} = 64\sqrt{3}\times $ height

$\therefore $ height $=$ $\displaystyle \dfrac{1920}{64}$
$= 30 \ cm$