Surface Areas and Volumes: 9 Math

NCERT Exercise 13.6 q5-8: 9th math

Surface Areas And Volumes Class nine Math NCERT Exercise 13.6  Question (5) It costs Rs 2200 to paint the inner curved surface of a cylindrical vessel 10 m deep. If the cost of painting is at the rate of Rs 20 per m², find.

(i) Inner curved surface area of the vessel,

(ii) radius of the base,

(iii) capacity of the vessel.

Solution

Given, Total cost of painting = Rs 2200

And, rate of painting = Rs 20 per m²

And, depth of the cylinder, i.e. height (h) = 10 m

Therefore, (i) Inner curved surface area of the vessel (ii) radius of the base and (iii) capacity of the vessel = ?

(i) Inner curved surface area of the vessel

Now, Curved Surface Area = Cost of painting/Rate of painting

= 2200/20

= 110 m²

Therefore, inner curved surface area of the given cylindrical vessel = 110 m² Answer

(ii) Radius of the base of the vessel

We know that, Curved surface area of a cylinder

= 2 ℼ r h

Therefore, curved surface area of the given cylinder

= 2 × `22/7` × r × 10 m

⇒ 110 m² = 2 × `22/7` r × 10 m

`=>r = (110m^2xx7)/(2xx22xx10m)`

⇒ r = 7/4 m = 1.75 m

Thus, radius of the given vessel

= 1.75 m Answer

(iii) Capacity of the vessel

Here, capacity of the vessel = volume of the vessel

Now, we know that Volume of a cylindrical vessel

= ℼ r^{2 h}

Thus, volume of the given cylindrical vessel

= `22/7` × (1.75 m)² × 10 m

= `22/7` × 1.75 m × 1.75 m × 10 m

= 22 × 0.25 m × 1.75 m × 10 m

= 5.5 m × 17.5 m²

= 96.25 m³ or 96.25 kl

Thus, volume of the given vessel

= 96.25 m³ or 96.25 kilo litre Answer

Surface Areas And Volumes Class nine Math NCERT Exercise 13.6  Question (6) The capacity of a closed cylindrical vessel of height 1 m is 15.4 litres. How many square metres of metal sheet would be needed to make it?

Solution

Calculation of volume of the cylinder

Given, Height of the closed cylindrical vessel = 1m

And, Capacity = 15.4 `l`

Thus, square meter of metal sheet to make it = ?

Here, capacity of the cylinder = volume of the cylinder

Here as given capacity of the cylinder

= 15.4 `l`

Now, since 1000 `l` = 1 m³

Therefore, `1\ l = (1 m^3)/1000`

Therefore, `15.4 \ l =1/1000xx15.4\ m^3`

= 0.0154 m³

Thus, volume of the given cylinder

= 0.015 m³

Calculation of the radius of the cylinder

Now, we know that Volume of a cylinder

= &8508; r² h

Therefore, volume of the given cylinder

= ℿ r² 1 m

⇒ 0.0154 m³ = `22/7` r² m

`=>r^2= (0.0154xx7)/22`

`=>r=sqrt(0.0049m^2)`

⇒ r = 0.07 m

Calculation of metal sheet required to make the given cylindrical vessel

Here, area of metal sheet required to make the given vessel = total surface area of the given cylindrical vessel.

Now, we know that, total surface area of a closed cylindrical vessel

= 2 ℿ r (r + h)

Thus, total surface area of the given closed cylindrical vessel

= 2 × `22/7` × 0.07 m (0.07 m + 1 m)

= 2 × 22 × 0.01 m × 1.07 m

= 44 × 0.0107 m²

=0.4708 m²

Thus, metal sheet required to make the given closed cylindrical vessel = 0.4708 m² Answer

Surface Areas And Volumes Class nine Math NCERT Exercise 13.6  Question (7) A lead pencil consists of a cylinder of wood with a solid cylinder of graphite filled in the interior. The diameter of the pencil is 7 mm and the diameter of the graphite is 1 mm. If the length of the pencil is 14 cm, find the volume of the wood and that of the graphite.

Solution

Given, Diameter of the pencil = 7 mm

Therefore, Radius of pencil (R)

= 7/2 mm = 3.5 mm

= 3.5/10 cm = 0.35 cm

Thus, radius of pencil = 0.35 cm

And, Diameter of graphite = 1mm

Therefore, Radius of the graphite (r)

= 1/2 mm

= 1/20 cm = 0.05 cm

Thus, radius of graphite (r) = 0.05 cm

And, Height of the pencil = 14 cm

Thus, volume of wood used in pencil and volume of graphite = ?

Calculation of volume of graphite

Here radius of graphite = 0.05 cm

And, height of the pencil, i.e. of graphite = 14 cm

Now, we know that, Volume of a cylinder

= ℼ r² h

Thus, volume of the graphite

= `22/7` × (0.05 cm)² × 14 cm

= 22 × 0.0025 cm² × 2 cm

= 22 × 0.0050 cm³

= 0.11 cm³

Thus, volume of the graphite used in given pencil

= 0.11 cm³

Calculation of volume of the wood used in pencil

Here, radius of pencil = 0.35 cm

and height = 14 cm

Now, we know that, Volume of a cylinder

= ℼ r² h

Therefore, Volume of pencil

= `22/7` × (0.35 cm)² × 14 cm

= 22 × 0.1225 cm² × 2 cm

= 5.39 cm³

Thus, volume of the pencil

= 5.39 cm³

Volume of wood = volume of pencil – volume of graphite

= 5.39 cm³ – 0.11 cm³

= 5.28 cm³

Thus, volume of wood used in pencil = 5.28 cm³

Thus, volume of wood used in pencil = 5.28 cm³ and volume of the graphite = 0.11 cm³ Answer

Surface Areas And Volumes Class nine Math NCERT Exercise 13.6  Question (8) A patient in a hospital is given soup daily in a cylindrical bowl of diameter 7 cm. if the bowl is filled with soup to a height of 4 cm, how much soup the hospital has to prepare daily to serve 250 patients?

Solution

Given, Diameter of the cylindrical bowl = 7 cm

Therefore, Radius of the cylindrical bowl = 7/2 cm = 3.5 cm

And, Height of the soup, i.e. height of the cylindrical bowl = 4 cm

And, total number of patient to whom soup is to be served = 250

Therefore, volume of soup to serve all the patient = ?

Calculation of Volume of the soup in one bowl

We know that, Volume of a Cylinder

= ℼ r² h

Therefore, volume of the soup in one bowl

= `22/7` × (3.5 cm)² 4 cm

= `22/7` × 3.5 cm × 3.5 cm × 4 cm

= 22 × 0.5 cm × 3.5 cm × 4 cm

= 22 × 1.75 cm² × 4 cm

= 22 × 7 cm³

= 154 cm³

Thus, volume of soup for one patient

= 154 cm³

Calculation of volume of soup for 250 patients

Now, since volume of soup for 1 patient = 154 cm³

Therefore, volume for 250 patients = 154 cm³ × 250

= 38500 cm³

Thus, volume of soup to be prepared by hospital daily = 38500 cm³ Answer

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