Surface Areas and Volumes Test

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Surface Areas and Volumes Test - 48

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Weekly Quiz Competition

Question 1
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Mohit painted the outside of a box of length $$2.5m$$, breadth $$2m$$ and height $$1m$$. How much area did he cover, if he painted all except the bottom of the box?

A
$$20m^2$$

B
$$16m^2$$

C
$$54m^2$$

D
$$14m^2$$

Solution

Area Mohit painted $$= 2(2.5\times 2+2\times 1+2.5\times 1)-(2.5\times 2)$$
$$=2(5+2+2.5)-(5)$$
$$=2(9.5)-5$$
$$=19-5$$
$$=14m^{2}$$
Question 2
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An aquarium is in the form of a cuboid whose external measures are $$80cm\times 30cm\times 40cm$$. The base side faces and back face are to be covered with a coloured paper. Find the area of the paper needed?

A
$$3000cm^2$$

B
$$5000cm^2$$

C
$$8000cm^2$$

D
$$7000cm^2$$

Solution

Area of paper required $$=$$ area of base $$+$$ 2 area of side $$+$$ area of back
$$ =(80 \times 30) + 2(30 \times 40)+(80 \times 40) $$
$$=2400+2400+3200 $$
$$=8000cm^{2}$$
Question 3
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The curved surface area of a cylinder of the base radius $$7\text{ cm}$$ and height $$25\text{ cm}$$.

A
$$1100\text{ cm}^3$$

B
$$1100\text{ cm}^2$$

C
$$1408\text{ cm}^3$$

D
$$1408\text{ cm}^2$$

Solution

$$\because$$ Curved Surface area of cylinder $$=2\pi rh$$
Given:
$$r=$$ radius $$=7\text{ cm}$$
$$h=$$ height $$=25\text{ cm}$$
$$\therefore$$ Curved Surface Area $$=2\times \cfrac{22}{7}\times 7\times 25$$ $$\text{cm}^{2}$$
$$=50\times 22$$ $$\text{cm}^{2}$$
$$\therefore$$ Curved Surface Area of cylinder $$=1100\text{ cm}^{2}$$
Question 4
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The volume of hollow sphere is

A
$$\frac { 4 }{ 3 } \pi \quad \left( { R }^{ 3 }-{ r }^{ 3 } \right) { cm }^{ 3 }$$

B
$$\frac { 2 }{ 3 } \pi \quad \left( { R }^{ 3 }-{ r }^{ 3 } \right) { cm }^{ 3 }$$

C
$$\frac { 1 }{ 2 } \pi \quad \left( { R }^{ 3 }-{ r }^{ 3 } \right) { cm }^{ 3 }$$

D
$$\frac { 5 }{ 6 } \pi \quad \left( { R }^{ 3 }-{ r }^{ 3 } \right) { cm }^{ 3 }$$

Solution

Where, $$R$$ is a radius of a sphere.
To find volume of a hollow sphere, we have to subtract the volume of the inner sphere from the volume of the outer sphere.
Let us assume the radius of outer sphere be $$R$$ and the radius of the inner radius be $$r.$$
The volume of a outer sphere $$=\dfrac{4}{3}\pi R^3\,cm^3$$

The volume of a inner sphere $$=\dfrac{4}{3}\pi r^3\,cm^3$$

$$\therefore$$ Volume of hollow sphere $$=$$ The volume of a outer sphere $$-$$ The volume of a inner sphere

$$=\dfrac{4}{3}\pi R^3-\dfrac{4}{3}\pi r^3$$

$$=\dfrac{4}{3}\pi(R^3-r^3)\,cm^3$$
Question 5
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A right circular cylinder tunnel of diameter $$2 \text{ m}$$ and length $$40 \text{ cm}$$ is to be constructed from a sheet of iron. The area of the sheet required in$$\text{ m}^2$$ is

A
$$\dfrac{4}{5}\pi$$

B
$$80\pi$$

C
$$160\pi$$

D
$$200\pi$$

Solution

Area of the required iron sheet $$=$$ Curved surface area of the cylinder
$$\because$$ Curved surface area of cylinder $$ = 2\pi rh $$
$$\implies $$ Area $$=2\pi\times \dfrac{40}{100}$$ $$[\because 1\text{ m}=100\text{ cm}$$ and $$r=\dfrac{D}{2}=\dfrac{2}{2}=1\text{ m}]$$
$$\implies $$ Area $$=\dfrac{4\pi}{5}\text{ m}^2$$
Question 6
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The diameter of the base of a right circular cylinder is $$21\mathrm { cm }$$ and its height is $$10\ \mathrm { cm }$$ . What is the area of the curved surface?

A
$$660\mathrm { cm } ^ { 2 }$$

B
$$930\mathrm { cm } ^ { 2 }$$

C
$$1320\mathrm { cm } ^ { 2 }$$

D
$$640\mathrm { cm } ^ { 2 }$$

Solution

$$Diameter =21\ cm$$
$$Radius=\dfrac{21}{2}\ cm$$
$$Height=10\ cm$$
We know that the curved surface area of the cylinder
$$=2\pi r h$$
$$=2\times \dfrac{22}{7}\times \dfrac{21}{2}\times 10$$
$$=220\times 3$$
$$=660\ cm^2$$
Question 7
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The diameter of a sphere whose surface area is $$346.5$$ $$\mathrm { cm } ^ { 2 }$$ is:

A
$$5.25$$ $$\mathrm { cm }$$

B
$$5.75$$ $$\mathrm { cm }$$

C
$$11.5$$ $$\mathrm { cm }$$

D
$$10.5$$ $$\mathrm { cm }$$

Solution

Let radius of sphere be $$r$$ cm.
Given, the surface area of a sphere $$=346.5\ \text{cm}^2$$.

We know, surface area of sphere $$=4\pi { r }^{ 2 }$$.

$$4\pi r^2 = 346.5 cm^2 $$

$$ { { r }^{ 2 } }=\cfrac { 346.5 }{ 4\pi } $$
$$=\dfrac{346.5\times7}{4\times22}$$
$$=\dfrac{2425.5}{4\times22}$$
$$=27.5625$$
$$\implies$$ $$ r=\sqrt { 27.5625 } =5.25\ \text{cm}$$ .

Therefore, option $$A$$ is correct.
Question 8
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If the surface area of sphere is $$(144\pi)\,{m}^{2}$$, then its volume is

A
$$(288\pi)\,{m}^{3}$$

B
$$(188\pi)\,{m}^{3}$$

C
$$(300\pi)\,{m}^{3}$$

D
$$(316\pi)\,{m}^{3}$$

Solution

Surface area of sphere $$=4\pi r^2$$
$$\Rightarrow 4\pi r^2=144 \pi$$

$$\Rightarrow r=\sqrt{\dfrac{144}{4}}=\sqrt{\left(\dfrac{12}{2}\right)^2}=6m$$
$$\Rightarrow $$ Volume $$=\dfrac{4}{3}\pi r^3=\dfrac{4}{3}\times \pi \times 6^3=288\pi m^3.$$

Hence, the answer is $$\left( 288\pi\right) m^3.$$
Question 9
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Calculate the surface area of a sphere with radius $$3.2 cm$$.

A
$$110.65 \ cm^2$$

B
$$128.61 \ cm^2$$

C
$$131.54 \ cm^2$$

D
None of these

Solution

Given, radius of the sphere, $$r=3.2 cm$$.

Therefore, the surface area of sphere $$=4\pi r^2$$
$$=4\pi (3.2)^2\ cm^2$$
$$=4 \times 3.14 \times 3.2 \times 3.2\ cm^2$$
$$=128.61 \ cm^2$$.

Hence, option $$B$$ is correct.
Question 10
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If $$'d'$$ is diameter of a sphere, then its volume is-

A
$$\dfrac{2}{3}\pi\ d^{3}$$

B
$$\dfrac{1}{6}\pi\ d^{3}$$

C
$$\dfrac{4}{3}\pi\ d^{3}$$

D
$$\dfrac{1}{24}\pi\ d^{3}$$

Solution

Volume of sphere:

$$V=\dfrac{4}{3}\pi r^3$$

$$r=\dfrac{d}{2}$$

$$\therefore V=\dfrac{1}{6}\pi d^3$$

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Re-Attempt Weekly Quiz Competition

Surface Areas and Volumes Test - 48

Result

Surface Areas and Volumes Test - 48

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SHARING IS CARING

Question 1

$$20m^2$$

$$16m^2$$

$$54m^2$$

$$14m^2$$

Solution

Question 2

$$3000cm^2$$

$$5000cm^2$$

$$8000cm^2$$

$$7000cm^2$$

Solution

Question 3

$$1100\text{ cm}^3$$

$$1100\text{ cm}^2$$

$$1408\text{ cm}^3$$

$$1408\text{ cm}^2$$

Solution

Question 4

$$\frac { 4 }{ 3 } \pi \quad \left( { R }^{ 3 }-{ r }^{ 3 } \right) { cm }^{ 3 }$$

$$\frac { 2 }{ 3 } \pi \quad \left( { R }^{ 3 }-{ r }^{ 3 } \right) { cm }^{ 3 }$$

$$\frac { 1 }{ 2 } \pi \quad \left( { R }^{ 3 }-{ r }^{ 3 } \right) { cm }^{ 3 }$$

$$\frac { 5 }{ 6 } \pi \quad \left( { R }^{ 3 }-{ r }^{ 3 } \right) { cm }^{ 3 }$$

Solution

Question 5

$$\dfrac{4}{5}\pi$$

$$80\pi$$

$$160\pi$$

$$200\pi$$

Solution

Question 6

$$660\mathrm { cm } ^ { 2 }$$

$$930\mathrm { cm } ^ { 2 }$$

$$1320\mathrm { cm } ^ { 2 }$$

$$640\mathrm { cm } ^ { 2 }$$

Solution

Question 7

$$5.25$$ $$\mathrm { cm }$$

$$5.75$$ $$\mathrm { cm }$$

$$11.5$$ $$\mathrm { cm }$$

$$10.5$$ $$\mathrm { cm }$$

Solution

Question 8

$$(288\pi)\,{m}^{3}$$

$$(188\pi)\,{m}^{3}$$

$$(300\pi)\,{m}^{3}$$

$$(316\pi)\,{m}^{3}$$

Solution

Question 9

$$110.65 \ cm^2$$

$$128.61 \ cm^2$$

$$131.54 \ cm^2$$

None of these

Solution

Question 10

$$\dfrac{2}{3}\pi\ d^{3}$$

$$\dfrac{1}{6}\pi\ d^{3}$$

$$\dfrac{4}{3}\pi\ d^{3}$$

$$\dfrac{1}{24}\pi\ d^{3}$$

Solution

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