Question

Assume π=22/7 unless stated otherwise.

1. The circumference of the base of a cylindrical vessel is 132 cm and its height is 25 cm. How many litres of water can it hold? (1 m3 = 1000)

2. The inner diameter of a cylindrical wooden pipe is 24 cm and its out diameter is 28 cm. The length of the pipe is 35 cm. Find the mass of the pipe, if 1cm3 of wood has a mass of 0.5 g.


3. A soft drink is available in two packs (i) a tin can with a rectangular base of length 5 cm and width 4 cm, having height of 15 cm
(ii) a plastic cylinder with circular base of diameter 7 cm and height 10 cm.
Which container has greater capacity and how much?



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Answer 1

1. Answer:

Here, the concept of volume of cylinder has been used. We see that we're given with the circumference of the base of cylindrical vessel and it's height. We've been asked to find it's volume. Since we're given with circumference we can find out it's radius by using circumference of circle formula. Then, equating all the values in volume of cylinder formula. We'll get the capacity of cylindrical vessel.

Formula used:

$\bullet\quad\boxed{\sf{Circumference\: of\: circle=2\times \pi\times r}}$

$\bullet\quad\boxed{\sf{Volume\: of\: cylinder=\pi\times {r}^{2}\times h}}$

Solution:

Given,

» Circumference of base = C = 132 cm

» Height of cylinder = h = 25 cm

• Let the radius of base of cylindrical vessel be r cm.

For radius of base of cylindrical vessel::

We know that,

$\\\longrightarrow\quad\sf{Circumference\: of\: circle=2\times \pi\times r}$

By applying values, we get:

$\\\longrightarrow\quad\sf{132=2\times\dfrac{22}{7}\times r}$

$\\\longrightarrow\quad\sf{\dfrac{132\times7}{2\times 22}=r}$

$\\\longrightarrow\quad\bold{r=21\: cm}$

• Here, we've calculated the radius of base of cylindrical vessel that is 21 cm.

Now, for volume of cylindrical vessel:

$\\\longmapsto\quad\sf{Volume\: of\: cylinder=\pi\times {r}^{2}\times h}$

By equating all the values, we get:

$\\\longmapsto\quad\sf{Volume\: of\: cylinder=\dfrac{22}{7}\times {21}^{2}\times 25}$

$\\\longmapsto\quad\underline{\boxed{\sf{Volume\: of\: cylinder=\bold{\purple{34649.37\: {cm}^{3}}}}}}$

Converting, cm³ to litres we get:

》 Volume of cylinder = $\sf{\dfrac{346949.37}{1000}}$

》 Volume of cylinder = $\sf{34.64937\: l}$

❝Hence, Volume of cylinder is 34.64937 litres.❞

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2. Answer:

Here, the concept of volume of concentric cylinder and it's mass has been used. We see that we're given with inner and outer diameter also length (height) of concentric cylinder is given. We've been asked to calculate it's mass. In order to find mass of cylinder we firstly need to find it's inner and outer radius. Then, Volume of concentric cylinder. After finding it's volume, multiply it's volume by mass of 1 cm³.

Formula used:

$\bullet\quad\boxed{\sf{Volume\: of\: concentric\: cylinder=\pi\times \left[{({r}_{2})}^{2}-{({r}_{1})}^{2}\right]\times h}}$

Solution:

Given,

» Inner diameter of pipe = 24 cm

» Outer diameter of pipe = 28 cm

» Height of the pipe = h = 35 cm

» Mass of 1 cm³ wood = 0.5 g

• Let inner radius be r1 and outer radius be r2.

》 Inner radius of pipe = r1 = 24/2 = 12 cm

》 Outer radius of pipe = r2 = 28/2 = 14 cm

Now, let's calculate it's volume:

$\\\dashrightarrow\quad\sf{Volume\: of\: pipe=\pi\times [{({r}_{2})}^{2}-{({r}_{1})}^{2}]\times h}$

By applying all the values:

$\\\dashrightarrow\quad\sf{Volume\: of\: pipe=\dfrac{22}{7}\times ({14}^{2}-{12}^{2})\times 35}$

$\\\dashrightarrow\quad\sf{Volume\: of\: pipe=\dfrac{22}{7}\times 35\times (196-144)}$

$\\\dashrightarrow\quad\sf{Volume\: of\: pipe=110\times 52}$

$\\\dashrightarrow\quad\sf{Volume\: of\: pipe= 5720\: {cm}^{3}}$

• Now, let's find it's mass:

$\\:\implies\quad\sf{Total\: mass= 0.5\times 5720}$

$\\:\implies\quad\underline{\boxed{\sf{Total\: mass = \bold{\red{2860\: g}}}}}$

❝Hence, mass of pipe is 2860 grams.❞

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3. Answer:

Here, the concept of volume of cuboid and volume of cylinder has been used. We're given with the dimensions of cuboid we've to find it's volume. Then, we're given with dimensions of cylinder (radius = diameter/2) and find it's volume. After calculating the volume of both container we have to calculate it's difference.

Formula used:

$\bullet\quad\boxed{\sf{Volume\: of\: cuboid=length\times width\times height}}$

$\bullet\quad\boxed{\sf{Volume\: of\: cylinder= \pi\times {r}^{2}\times h}}$

Solution:

(i) Given,

» Length of cuboid = L = 5 cm

» Width of cuboid = W = 4 cm

» Height of cuboid = H = 15 cm

We know,

$\\\longrightarrow\quad\sf{Volume\: of\: tin= L\times W\times H}$

Substituting all the values;

$\\\longrightarrow\quad\sf{Volume\: of\: tin=5\times 4\times 15}$

$\\\longrightarrow\quad\underline{\boxed{\sf{Volume\: of\: tin= \bold{\green{300\: {cm}^{3}}}}}}$

Now,

» Radius of cylinder = r = d/2 = 7/2 cm

» Height of cylinder = h = 10 cm

We know,

$\\\dashrightarrow\quad\sf{Volume\: of\: cylinder=\pi\times {r}^{2}\times h}$

$\\\dashrightarrow\quad\sf{Volume\: of\: cylinder=\dfrac{22}{7}\times{\left(\dfrac{7}{2}\right)}^{2}}$

$\\\dashrightarrow\quad\underline{\boxed{\sf{Volume\: of\: cylinder= \bold{\green{385\: {cm}^{3}}}}}}\;\bigstar$

• Here, Volume of cylinder has greater capacity than volume of tin.

》 Difference = 385 - 300 = 85 cm³

❝Hence, Cylinder container has greater capacity by 85 cm³.❞