## RS Aggarwal Class 9 Ex 13A Chapter 13

**Q.1:** **Find the volume and curved surface area of a right circular cylinder of height 21 cm and base radius 5 cm.**

**Solution:**

Here, r = 5 cm and h = 21 cm

∴ Volume of the cylinder = 𝜋r^{2}h = ^{3}

∴ Curved surface area of the cylinder = 2𝜋rh = ^{2}

**Q.2:** **The diameter of a cylinder is 28 cm and its height is 40 cm. Find the curved surface area, total surface area and the volume of the cylinder.**

**Solution:**

Here, diameter = 28 cm, Radius = 14 cm and Height = 40 cm

∴ Curved surface area = 2𝜋rh = ^{2}

∴ Total surface area = 2𝜋rh + 2𝜋r^{2} =

= (3520 + 1232) = 4752 cm^{2 }

∴ Volume of the cylinder = 𝜋r^{2}h = ^{3}

**Q.3:** **Find the weight of a solid cylinder of radius 10.5 cm and height 60 cm if the material of the cylinder weighs 5 g per cm ^{3}.**

**Solution:**

Here, radius(r) = 10.5 cm and height = 60 cm

∴ Volume of the cylinder = 𝜋r^{2}h = ^{3}

∴ Weight of the solid cylinder if the material of the cylinder weighs 5 g per cm^{3} = (20790)(5) = 103950 g =

**Q.4:** **The curved surface area of a cylinder is 1210 cm ^{2} and its diameter is 20 cm. Find its height and volume.**

**Solution:**

Here, curved surface area = 1210 cm^{2}

Diameter = 20 cm

∴ Curved surface area of the cylinder = 2𝜋rh

∴ Height = 19.25 cm

∴ Volume of the cylinder = 𝜋r^{2}h = ^{3}

∴ Volume of the cylinder = 6050 cm^{3}

**Q.5:** **The curved surface area of a cylinder is 4400 cm ^{2} and the circumference of its base is 110 cm. Find the height and volume of the cylinder.**

**Solution:**

Let base radius be ‘r’ and height be ‘h’

Then, 2𝜋rh = 4400 cm^{2}

And, 2𝜋r = 110 cm

∴

∴ Volume of the cylinder = 𝜋r^{2}h = ^{3 }

**Q.6:** **The radius of the base and height of a cylinder are in the ratio 2 : 3. If its volume is 1617 cm ^{3}, find the total surface area of the cylinder.**

**Solution:**

Let the radius ® = 2x cm and height (h) = 3x cm

Then, Volume of cylinder = 𝜋r^{2}h

Volume =

Volume =

Volume =

But, given volume = 1617 cm^{3}

Thus solving the equation for x, we get,

∴ Radius = 2x = 7 cm

And height = 3x = 20.5 cm

Total surface area = 2𝜋r(h + r) = ^{2} = ^{2} = 770 cm^{2 }

**Q.7:** **The total surface area of a cylinder is 462 cm ^{2}. Its curved surface area is one – third of its total surface area. Find the volume of the cylinder.**

**Solution:**

Curved surface area =

=

Total surface area – Curved surface area = (462 – 154) cm^{2} = 308 cm^{2}

Now, curved surface area = 2𝜋rh = 154 cm^{2} =

Now, r = 7 cm and h-= 3.5 cm

∴ Volume of the cylinder = 𝜋r^{2}h = ^{3 }

∴ Volume of the cylinder = 539 cm^{3}

**Q.8:** **The total surface area of a solid cylinder is 231 cm ^{2} and its curved surface area is (⅔) of the total surface area. Find the volume of the cylinder.**

**Solution:**

Curved surface area =

=

(Total surface area) – (Curved surface area) = (231 – 154) = 77 cm^{2}

2𝜋r^{2} = 77 cm^{2}

Now, 2𝜋rh = 154 cm^{2}

Now,

Volume of the cylinder = 𝜋r^{2}h = ^{3 }

Therefore, Volume of the cylinder = 269.5 cm^{3}

**Q.9:** **The sum of the height and radius of the base of a solid cylinder is 37 m. if the total surface area of the cylinder is 1628 m ^{2}, find its volume.**

**Solution:**

Here, (r + h) = 37 m [given]

And total surface area = 2𝜋r(r + h) = 1628 m^{2}

^{2}

And, (r + h) = 37 m

^{2}h = ^{3 }

**Q.10:** **The ratio between the curved surface area and the total surface area of a right circular cylinder is 1 : 2. Find the volume of the cylinder, if its total surface area is 616 cm ^{2}.**

**Solution:**

Curved surface area = 2𝜋rh

Total surface area = 2𝜋r(h + r)

Since they are in the ratio of 1:2

2𝜋r(h + r) = 616 cm^{2}

^{2} = 616 cm^{2} [putting h = r]

Then, r = 7 cm and h = 7 cm

Therefore, Volume of the cylinder = 𝜋r^{2}h =^{3 }

Volume of the cylinder = 1078 cm^{3}

**Q.11:** **1 cm ^{3} of gold drawn into a wire 0.1 mm in diameter. Find the length of the wire.**

**Solution:**

1 cm^{3} = 1 cm × 1 cm × 1 cm

Where, 1 cm = 0.01 m

Therefore, Volume of the gold = 0.01 m × 0.01 m × 0.01 m = 0.000001 m^{3} . . . . . . . (1)

Diameter of the wire drawn = 0.1 mm

Radius of the wire drawn = 0.05 mm = 0.00005 m . . . . . . (2)

Length of the wire = ‘h’ m . . . . . . (3)

Volume of the wire drawn = Volume of the gold

^{2}h = 0.000001

∴ The length of the wire is 127.27 m

**Q.12:** **The radii of two cylinders are in the ratio 2 : 3 and their heights are in the ratio 5 : 3. Calculate the ratio of their volumes and the ratio of their curved surfaces.**

**Solution:**

Let the radii of the two cylinders be 2R and 3R.

And their heights be 5H and 3H

Then,

∴ the ratio of their volumes = 20:27

Now,

∴ The ratio of their curved surface = 10:9

**Q.13:** **A powder of tin has a square base with side 12 cm and height 17.5 cm. Another is cylindrical with diameter of its base 12 cm and height 17.5 am. Which has more capacity and by how much?**

**Solution:**

For the tin having square base,

Side = 12 cm and height = 17.5 cm.

∴ Volume = (12 × 12 × 17.5) cm^{3 }= 2520 cm^{3}

Now, diameter of tin with cylindrical base = 12 cm

∴ Radius = 6 cm and height = 17.5 cm.

∴ Volume =

Tin with square base has more capacity by (2520 – 1980) cm^{3} = 540 cm^{3 }

**Q.14:** **A cylindrical bucket, 28 cm in diameter and 72 cm high, is full of water. The water is emptied into a rectangular tank, 66 cm long and 28 cm wide. Find the height of the water level in the tank.**

**Solution:**

Here, cylindrical bucket has diameter = 28 cm

∴ Radius = 14 cm and height = 72 cm

Length of the tank = 66 cm

Breadth of the tank = 28 cm

∴ Volume of the tank = Volume of cylindrical bucket.

^{2}h

∴ The height of the water level in the tank = 24 cm.

**Q.15:** **If 1 cm ^{3} of cast iron weighs 21 g, find the weight of a cat iron pipe of length 1 m with a bore of 3 cm in which the thickness of the metal is 1 cm.**

**Solution:**

Internal radius = (3/2) = 1.5 cm

And, external radius = (1.5 + 1) cm = 2.5 cm

Volume of cast iron = [𝜋 × (2.5)^{2} × 100 – 𝜋 × (1.5)^{2} × 100] cm^{3} = 𝜋 × 100 × [(2.5)^{2} – (1.5)^{2}] cm^{3 }

=

=

Weight =

(Because, 1kg = 1000g)

Weight = 26.4 kg.

Hence, the weight of the iron pipe = 26.4 kg

**Q.16:** **A cylindrical tube, open at both ends, is made of metal. The internal diameter of the tube is 10.4 cm and its length is 25 cm. The thickness of the metal is 8 mm everywhere. Calculate the volume of the metal.**

**Solution:**

Internal diameter of the tube = 10.4 cm

Internal radius = 5.2 cm

And length = 25 cm

And external radius = (5.2 + 0.8) = 6 cm

Required volume =

=

=

= ^{3 }

Therefore, the volume of the metal = 704 cm^{3 }

**Q.17:** **The barrel of a fountain pen, cylindrical in shape, is 7 cm long and 5 mm in diameter. A full barrel of ink in the pen will be used up on writing 330 words on an average. How many words would use up a bottle of ink containing one – fifth of a liter?**

**Solution:**

Length = 7 cm = height

Diameter = 5 mm

Volume of the barrel = 𝜋r^{2}h = ^{3}

=

Hence,

So,

**Q.18:** **A lead pencil consists of a cylinder of wood with a solid cylinder of graphite fitted into it. The diameter of the pencil is 7 mm, the diameter of the graphite is 1 mm and the length of the pencil is 10 cm. Calculate the weight of the whole pencil, if the specific gravity of the wood is 0.7 g/ cm ^{3} and that of the graphite is 2.1 g/cm^{3}.**

**Solution:**

Weight of the graphite =

Weight of wood =

=

Therefore, total weight of the pencil =

=

Therefore, weight of the whole pencil = 2.805 g