Perimeter, area and volume - class-VIII

 Description: perimeter, area and volume Number of Questions: 15 Created by: Anumati Koshy Tags: perimeter, area and volume solids measures and the circle surface areas and volumes maths mensuration
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From a solid sphere of radius $R$, a concentric solid sphere of radius $\dfrac{R}{2}$ is removed. The total surface area increases by

1. $0\%$

2. $25\%$

3. $50\%$

4. $75\%$

Correct Option: B
Explanation:
Solution:- (B) $25 \%$
Initial area of sphere $\left( {A} _{1} \right) = 4 \pi {R}^{2}$
New area of sphere $\left( {A} _{2} \right) = 4 \pi {R}^{2} + 4 \pi {\left( \cfrac{R}{2} \right)}^{2} = 5 \pi {R}^{2}$
$\therefore$ Increase in area $= \cfrac{{A} _{2} - {A} _{1}}{{A} _{1}} \times 100$
$\Rightarrow$ Increase in area $= \cfrac{5 \pi {R}^{2} - 4 \pi {R}^{2}}{4 \pi {R}^{2}} \times 100 = 25 \%$
Hence the area will be increased by $25 \%$.

The volume of triangular prism whose adjacent sides are $\bar a,\ \bar b,\ \bar c$ each of magnitude $4$ units and each is inclined at an angel $\dfrac {\pi}{3}$ with other two is

1. $16\sqrt {2}$

2. $16\sqrt {3}$

3. $8\sqrt {3}$

4. $8\sqrt {2}$

Correct Option: B

If angle of minimum deviation through an equilateral prism is $40^o$, angle of incidence (being equal to angle of emergence) would be

1. $50^o$

2. $60^o$

3. $40^o$

4. none of these

Correct Option: A
Explanation:
For minimum deviation $i=ei=e$
$a+δm=i+ea+δm=i+e$
$a+δm=2ia+δm=2i$
$60^o+40^o=2i$
$i=50^o$

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

1. $120 cm^3.$

2. $240 cm^3.$

3. $640 cm^3.$

4. $900 cm^3.$

Correct Option: C
Explanation:
Given that:
Side $a=8\ cm$, Height $h=30\ cm$
As we know that
Volume of regular square pyramid
$\Rightarrow a^2\dfrac{h}{3}$
$\Rightarrow 8^2\dfrac{30}{3}$
$\Rightarrow 64\times 10$
$\Rightarrow 640\ cm^3$
This is the required solution.

A square pyramid can contain $16\ m^3$ of water. The height of the pyramid is $3\ m$. Calculate the length of base of the square pyramid.

1. $16\ m$

2. $12\ m$

3. $4\ m$

4. $2\ m$

Correct Option: C
Explanation:

Volume of square pyramid  $=16{ m }^{ 3 }$

$\Rightarrow \quad \dfrac { 1 }{ 3 } \times { a }^{ 2 }\times h=16{ m }^{ 3 }\Rightarrow \dfrac { 1 }{ 3 } \times { a }^{ 2 }\times 3=16\Rightarrow a=\sqrt { 16 } =4cm$
$\therefore$  Length of base $= 4m$

$VPQRS$ is rectangle based pyramid where $PQ = 30\ cm, QR = 20\ cm$ and volume is $2000\ {cm}^3$, then height (in cm) is

1. $20$

2. $40$

3. $10$

4. $30$

Correct Option: C
Explanation:

Given : Length of base$(l)=30\ cm$, width of base$(h)=20\ cm$, Volume of pyramid$=2000\ cm^3$

Let $h$ be the height of the pyramid
We know that, volume of pyramid $=\dfrac{l\times w\times h}{3}$
$\implies 2000\ cm^3 = \dfrac{30 cm\times 20 cm\times h}{3}$
$\implies h=\dfrac{2000\times 3}{30\times 20} cm=10 cm$
Hence, height is $10 cm$.

State whether true or false :

If $s$ is the perimeter of the base of a prism, $n$ is the number of sides of the base, $S$ is the total length of the edges and $h$ is the height, then $S=nh+2s$.

1. True

2. False

Correct Option: A
Explanation:

True. Total length of Prism $= S$
Length of all the heights $= nh$
Perimeter of base and head $= 2s$
$\therefore S = nh + 2s$

The base of right prism is a triangle whose perimeter is 28 cm and the inradius of the triangle is 4 cm. If the volume of the prism is 366 cc, then its height is

1. 6.54 cm

2. 8 cm

3. 4 cm

4. None of these

Correct Option: A
Explanation:

Perimeter of triangle is $2{s}=28\ cm\implies s=14$

Inradius of triangle $r=4\implies \Delta=r.s=56$
Volume of prism $=366$ cc
$\implies$ Area of triangle $\times$ height $=366$
Height $=\dfrac{366}{56}=6.54$

For a prism, $A = 60^o$, $\mu = \sqrt{\dfrac{7}{3}}$, then the minimum possible angle of incidence, so that the light ray is refracted from the second surface is

1. $30^o$

2. $60^o$

3. $90^o$

4. $40^o$

Correct Option: A

The slant height of a right pyramid having square base of side $10cm$ and vertical height $15cm$ is

1. $5\sqrt{10}cm$

2. $6\sqrt{10}cm$

3. $7\sqrt{10}cm$

4. $8\sqrt{10}cm$

Correct Option: A

Consider an incomplete pyramid of balls on a square base having $18$ players, and having $13$ balls on each side of the top layer. Then the total number $N$ of balls in that pyramid satisfies

1. $9000 < N <10000$

2. $8000 < N < 9000$

3. $7000 < N < 8000$

4. $10000 < N < 12000$

Correct Option: B
Explanation:

Top layer has $(13 \times 13)$ balls
Similarly one layer below top layer will have $(14 \times 14)$ balls and we have $18$ lesens to total number of ball
$N = (13)^2 + (14)^2 + . . . . + (30)^2$

$\displaystyle N = \frac {30 \times 31 \times 61} {6} = \frac {12 \times 13 \times 25} {6}$

$N = 8805$

The circumference of a 1 cm thick pipe is 44 cm. The level of water that 7 cm of pipe can hold is

1. $798 cm^3$

2. $308 cm^3$

3. $792 cm^3$

4. $795 cm^3$

Correct Option: C
Explanation:

Given that

$2\pi r=44$
$r=7\ cm$

So, the inner radius of the pipe $=7-1=6\ cm$

Therefore, the volume of the pipe
$=\pi r^2h$
$=\pi \times 6^2\times 7$
$=792\ cm^3$

Hence, this is the answer.

The base of a right prism is a square of perimeter 20 cm and its height is 30 cm. The volume of the prism is

1. $700 cm^3$

2. $750 cm^3$

3. $800 cm^3$

4. $850 cm^3$

Correct Option: B
Explanation:

Given, perimeter $=4a=20$cm
$\therefore a=5$ cm
Area $=a^2=25 cm^2$
Volume $=$ Area $\times$ Height
Volume $=25 \times 30$
Volume $=750 cm^3$

The base of a right prism is an equilateral triangle of edge $12$m. If the volume of the prism is $288\sqrt 3m^3$, then its height is:

1. $6$m

2. $8$m

3. $10$m

4. $12$m

Correct Option: B
Explanation:

length of Equilateral triangle $= 12 m$
Area of equilateral triangle = $\displaystyle \frac{\sqrt{3}}{4}a^2$ = $\displaystyle \frac{\sqrt{3}}{4}(12)^2$ = $36\sqrt{3}$
Volume of prism = $288\sqrt{3} m^3$ = Area of triangle X height
$288\sqrt{3} m^3$ = $36 \sqrt{3} \times$ height
$\therefore$ Height $= 8 m$

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