Surface Areas and Volumes: 9 Math

NCERT Exemplar Exercise 13.4 Q5-8: 9th math

Solution of NCERT Exemplar Exercise 13.4 Surface Areas And Volumes Class 9 Math   Question (5) The volumes of the two spheres are in the ratio 64 : 27. Find the ratio of their surface areas.

Solution

Given, ratio of volumes of two spheres = 64 : 27

Thus, ratio of surface areas of the given two spheres = ?

Now, we know that, Volume of sphere = 4/3 ℼ r³

Thus, ratio of volumes of two spheres having radii R and r

= 4/3 ℼ R³ : 4/3 ℼ r³

= R³ : r³

Thus, ratio of volume of two spheres = ratio of cube of their radii

⇒ 64 : 27 = (4)³ : (3)³

Thus, Thus, radius of big sphere (R) = 4

And, radius of small sphere (r) = 3

Now, we know that, surface area of sphere = 4 ℼ r²

Thus, ratio of surface area of two spheres having radii R and r respectively

= 4 ℼ R² : 4 ℼ r²

= R² : r²

Thus, ratio of surface area of two spheres = ratio of square of their radii

Thus, ratio of curved surface area of given two spheres

= 4² : 3²

= 16 : 9

Thus, ratio of surface areas of given two spheres = 16 : 9 Answer

Solution of NCERT Exemplar Exercise 13.4 Surface Areas And Volumes Class 9 Math   Question (6) A cube of side 4 cm contains a sphere touching its sides. Find the volume of the gap in between.

Solution

Given, side of cube = 4 cm

And, this cube contains a sphere touching its side,

Thus, volume of gap in between the cube and sphere = ?

Here, volume of gap in between the cube and sphere = volume of cube – volume of sphere

Calculation of the volume of the cube

We know that, Volume of a cube = (side)³

Thus, volume of the given cube = (4 cm)³

= 64 cm³

Thus, volume of the cube = 64 cm³

Calculation of volume of given sphere

Now, since sphere kept in the cube is touching its side

Thus, diameter of sphere = edge of cube = 4 cm

Thus, radius of the sphere = 4/2 = 2 cm

Now, we know that, Volume of a sphere = 4/3 ℼ r³

Thus, volume of the given cube = 4/3 ℼ (2 cm)³

= 4/3 × 22/7 × 8 cm³

= 88/21 × 8 cm³

= 33.524 cm³

Thus, volume of the sphere = 33.524 cm³

Calculation of volume of gap between sphere and the edges of cube

The volume of gap between sphere and edges of the cube = volume of cube – volume of sphere

= 64 cm³ – 33.524 cm³

= 30.476 cm³

Thus, volume gap between sphere and cube = 30.476 cm³ Answer

Solution of NCERT Exemplar Exercise 13.4 Surface Areas And Volumes Class 9 Math   Question (7) A sphere and a right circular cylinder of the same radius have equal volumes. By what percentage does the diameter of the cylinder exceed its height?

Solution

Given, the radius and volume are equal of a sphere and a right circular cylinder.

Thus, percent of exceed of diameter of the height of the cylinder = ?

Let, radius of sphere and that of right circular cylinder = r

And, height of the right circular cylinder = h

Now, we know that, Volume of a sphere = 4/3 ℼ r³

And, Volume of a right circular cylinder = ℼ r² h

Now, according to question,

Volume of sphere = volume of right circular cylinder

⇒ 4/3 ℼ r³ = ℼ r² h

⇒ 4/3 r = h

⇒ r = 3/4 h

Thus, radius of the cylinder = 3/4 h

Thus, diameter of the cylinder `=2xx3/4h`

⇒ Diameter of the cylinder = 3/2 h

⇒ Diameter of the cylinder = 1.5 h

Thus, for diameter increase in height = 1.5 h – h = 0.5 h

Now, in h the increase = 0.5 h

Thus, in 1, the increase `=(0.5h)/h=0.5`

Thus, in 100, the increase = 0.5 × 100 = 50%

Thus, diameter of the cylinder exceeds from its height = 50% Answer

Solution of NCERT Exemplar Exercise 13.4 Surface Areas And Volumes Class 9 Math   Question (8) 30 circular plates, each of radius 14 cm and thickness 3 cm are placed one above the another to form a cylindrical solid. Find:

(i) the total surface area

(ii) volume of the cylinder so formed.

Solution

Given, thickness of one circular plate = 3 cm

And, radius of circular plate = 14 cm

And total number of plates place one above another to form a cylindrical solid = 30

Thus, the total surface area and volume of the cylinder formed = ?

(i) The total surface area of the cylindrical solid formed

Here, since, total 30 circular plates area kept above one another and thickness of one plate = 3 cm

Thus, total height of the cylinder so formed

= thickness of one plate × total number of plate

= 3 cm × 30 = 90 cm

And, radius of the plate = 14 cm

Now we know that, total surface area of a cylinder

= 2 ℼ r (r + h)

Thus, total surface area of the given cylinder

= 2 × 22/7 × 14 cm × (14 cm + 90 cm)

= 44 × 2 cm × 104 cm

= 44 × 208 cm²

= 9152 cm²

Thus, total surface area of the cylindrical solid formed

= 9152 cm² Answer

(ii) The volume of the cylinder so formed

We know that, volume of a cylinder

= ℼ r² h

= 22/7 × (14 cm)² × 90 cm

= 22/7 × 14 cm × 14 cm × 90 cm

= 22 × 2 cm × 14 cm × 90 cm

= 22 × 2520 cm³

= 55440 cm³

Thus, volume of the cylinder formed = 55440 cm³ Answer

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