3 Marks Question

Question 13 Marks

Metal spheres, each of radius $2\ cm$, are packed into a rectangular box of internal dimensions $16cm \times 8cm \times 8cm.$ When $16$ spheres are packed the box is filled with preservative liquid. Find the volume of this liquid. Give your answer to the nearest integer. $[\text{Use }\pi=3.14]$

Answer

Internal volume of a rectangular box$ = 16\ cm \times 8\ cm \times 8\ cm =1024\ cm^2$
Volume of a sphere $=\frac{4}{3}\pi\text{r}^3$
$=\frac{4}{3}\times3.14\times(2)^3=\frac{100.48}{3}=33.49\text{cm}^3$
Volume of $16$ such spheres $=(33.49\times16)\text{cm}^3=535.84\text{cm}^3$
Where $16$ spheres are packed, the box is filled with preservative liquid.
Volume of the preservative liquid $=1024-535.84\text{cm}^3=488.16\text{cm}^3$

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Question 23 Marks

A school provides milk to the students daily in a cylindrical glasses of diameter $7\ cm$. If the glass is filled with milk upto an height of $12\ cm$, find how many litres of milk is needed to serve $1600$ students.

Answer

Given, diameter of glass = 7cmRadius of glass, $\text{r}=\frac{7}{2}\text{cm}$
$\therefore$ Milk contained in the cylinderical glass = Volume of cylindrical glass
$=\pi\text{r}^2\text{h}=\frac{22}{7}\times\frac{7}{2}\times\frac{7}{2}\times12=462\text{cm}^3$
Now, Milk required for 1600 students = 462 \times 1600 = 739200cm$^3$
$=\frac{73920}{100}=7392\text{L}\ \ \ \Big[\because1\text{cm}^3=\frac{1}{1000}\text{L}\Big]$
Hence, $739.2L$ milk is needed to serve $1600$ students.

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Question 33 Marks

A cylindrical roller $2.5m$ in length, $1.75m$ in radius when rolled on a road was found to cover the area of $5500m^2$. How many revolutions did it make?

Answer

Length (height) of cylinder roller is $2.5\ m$ and radius of the roller is $1.75\ m$.In one revolution area covered = lateral surface area of the cylinder
$2\pi\text{r}\text{h}=2\times\frac{22}{7}\times1.75\times2.5\text{m}^2$
$44\times0.25\times2.5=27.5\text{m}^2$
Total area on the road covered by cylinder roller $= 5500m^2$.
Hence, number of revolution made by the roller
$=\frac{\text{Total area covered}}{\text{Area covered in one revolution}}$
$=\frac{5500}{27.5}=200\text{ revolutions}.$

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Question 43 Marks

A right triangle with sides $6\ cm, 8\ cm$ and $10\ cm$ is revolved about the side $8\ cm$. Find the volume and the curved surface of the solid so formed.

Answer

When a right triangle with sides $6 cm, 8 cm$ and $10 \ cm $is revolved about the side $8 \ cm ,$
then solid formed is a cone whose height of a cone,
$h =8 cm$ and radius of a cone, $r =6 cm$.
Slant height of a cone, $I =10 cm$ Volume of a cone
$=\frac{1}{3}\pi\text{r}^2\text{h}=\Big(\frac{1}{3}\Big)\times\Big(\frac{22}{7}\Big)\times6\times6\times8$
$\Rightarrow\frac{6336}{21}=301.7\text{cm}^3$ And curved surface of the area of cone $=\pi\text{rl}$
$\Rightarrow\Big(\frac{22}{7}\Big)\times6\times10=\frac{1320}{7}=188.5\text{cm}^2$
Hence, the volume and surface area of a cone are $301.7cm^3 $ and $188.5cm^2$, respectively.

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Question 53 Marks

A small village, having a population of $5000$ , requires $75$ litres of water per head per day. The village has got an overhead tank of measurement $40 m \times 25 \ m \times 15 \ m$. For how many days will the water of this tank last?

Answer

Given, total population of a small village $=5000$ Water required per head per day $=75 L$
Volume of water required for a small village per day $=5000 \times 75=375000 L$
$=\frac{375000}{1000}\text{m}^3=375\text{m}^3\ [\because1\text{m}^3=1000\text{L}]$
Total capacity of water in overhead tank = Volume of overhead tank
$=40\times25\times15=15000\text{m}^3$
$\therefore\ \text{Number of day}=\frac{\text{Total capacity of water in over speed tank}}{\text{Volume of water required for small village per day}}$
$=\frac{1500}{375}=40\text{ day}$
Hence, water of this tank will be last in $40$ days.

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Question 63 Marks

A storage tank is in the form of a cube. When it is full of water, the volume of water is $15.625 m^3$. If the present depth of water is $1.3 \ m$ , find the volume of water already used from the tank.

Answer

Let side of a cube be $=X m . \therefore$ Volume of cubical tank $=15.625 m^3$ [given] $\Rightarrow X^3=15.625 m^3$ $\Rightarrow X =2.5 m$
and present depth of water in cubical tank $=1.3 m \therefore$ Height of water used $=2.5-1.3 m=1.2 m$ Now, volume of water used $=1.2 \times 2.5 \times 2.5=7.5 m^3 7.5 \times 1000=7500 L\left[\therefore 1 m^3=1000 L\right]$ Hence, the volume of water already used from the tank is 7500 L .

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Question 73 Marks

Find the amount of water displaced by a solid spherical ball of diameter $4.2\ cm$, when it is completely immersed in water.

Answer

Given: Diameter of spherical ball $= 4.2\ cm$
Then Radius of spherical ball $(r) = 2.1\ cm$
Amount of water displaced by a solid spherical ball = Volume of solid spherical boll.
Volume of spherical ball $=\frac{4}{3}\pi\text{r}^3=\frac{4}{3}\times\frac{22}{7}\times(2.1)^3$
$=\frac{88}{21}\times\frac{21}{10}\times\frac{21}{10}\times\frac{21}{10}=38.808\text{cm}^3$
Hence, the amount of water displaced by solid spherical boll when it completely immersed in water is $38.808cm^3$.

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