3 Marks Question

Question 13 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 40m × 25m × 15m. For how many days will the water of this tank last?

Answer

Given, total population of a small village = 5000Water required per head per day = 75L
Volume of water required for a small village per day = 5000 × 75 = 375000L
$=\frac{375000}{1000}\text{m}^3=375\text{m}^3\ [\because1\text{m}^3=1000\text{L}]$
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 23 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 $38.808\ cm^3.$

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Question 33 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$, l = 10\ cm$
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 $301.7\ cm^3$ and $188.5\ cm^2,$ respectively.

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Question 43 Marks

Metal spheres, each of radius $2\ cm$, are packed into a rectangular box of internal dimensions $16\ cm \times 8\ cm \times 8\ cm.$ 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 $
$=1024\ cm^3$
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 53 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 $= Xm$.
$\therefore$ 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 $= 1.2 \times 2.5 \times 2.5$
$= 7.5\ m^3 7. 5 \times 1000 = 7500\ L$
$[\therefore1\text{ m}^3=1000\text{ L]}$
Hence, the volume of water already used from the tank is $7500\ L.$

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Question 63 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 $= 7\ cm$ Radius 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 $
$= 739200\ cm^3$
$=\frac{73920}{100}$
$=7392\text{ L}\ \ \ \Big[\because1\text{ cm}^3=\frac{1}{1000}\text{ L}\Big]$
Hence, $739.2\ L$ milk is needed to serve $1600$ students.

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Question 73 Marks

A cylindrical roller $2.5\ m$ in length, $1.75\ m$ in radius when rolled on a road was found to cover the area of $5500\ m^2.$ 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
$=\frac{\text{Total area covered}}{\text{Area covered in one revolution}}$
$=\frac{5500}{27.5}$
$=200$ revolutions

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