Surface Areas and Volumes: 9 Math
NCERT Exercise 13.7: 9th math
Important Formula
Volume of a Right Circular Cone `=1/3\ pi\ r^2\ h`
Slant height of a Right Circular Cone (`l`) `=sqrt(h^2+r^2)`
Where, `l` = slant height, r = radius of base and h = height of the cone.
NCERT Exercise 13.7 Questions and Answers
Assume `pi=22/7`, unless stated otherwise
Surface Areas And Volumes Class nine Math NCERT Exercise 13.7 Question (1) Find the volume of the right circular cone with
(i) Radius 6 cm, height 7 cm
(ii) Radius 3.5 cm, height 12 cm
Solution
(i) Radius 6 cm, height 7 cm
Given, radius of a right circular cone = 6 cm
And, Height of the right circular cone = 7 cm
Therefore, volume of the given right circular cone = ?
We know that Volume of a Right Circular Cone `=1/3\ pi\ r^2\ h`
Therefore, volume of the given right circular cone
= `1/3xx22/7` × 6 cm × 6 cm × 7 cm
= 22 × 2 cm × 6 cm × 1 cm
= 44 cm × 6 cm2
= 264 cm3
Thus, volume of the given right circular cone = 264 cm3 Answer
(ii) radius = 3.5 cm, height = 12 cm
Given, radius of the right circular cone = 3.5 cm
And, height of the right circular cone = 12 cm
Therefore, volume of the given right circular cone = ?
We know that Volume of a Right Circular Cone `=1/3\ pi\ r^2\ h`
Therefore, volume of the given right circular cone
= `1/3xx22/7` × (3.5 cm)2 × 12 cm
= `1/3xx22/7` × 3.5 cm × 3.5 cm × 12
= 22 × 0.5 cm × 3.5 cm × 4 cm
= 22 × 1.75 cm2 × 4 cm
= 22 × 7 cm3
= 154cm3
Thus, volume of the given right circular cone
= 154 cm3 Answer
Surface Areas And Volumes Class nine Math NCERT Exercise 13.7 Question (2) Find the capacity in litres of a conical vessel with
(i) radius 7 cm, slant height 25 cm
(ii) height 12 cm, slant height 13 cm
Solution(i)
(i) radius 7 cm, slant height 25 cm
Given, Radius of the right circular cone = 7 cm
And, Slant height of the right circular cone = 25 cm
Therefore, capacity, i.e. volume of the given right circular cone = ?
We know that Volume of a Right Circular Cone `=1/3\ pi\ r^2\ h`
Therefore, it is necessary to calculate height before calculation of volume of the given right circular cone.
Here, we know that, slant height, radius and height of a right circular cone together form a right angled triangle.
Therefore,
[slant height of a right circular cone (`l`)]2
= (height)2 + (radius)2
Thus, slant height of given right circular cone,
(25 cm)2 = (h)2 + (7 cm)2
⇒ 625 cm2 = h2 + 49 cm2
⇒ h2 = 625 cm2 – 49 cm2
⇒ h2 = 576 cm2
Therefore, `h=sqrt(576 cm^2)`
⇒ h = 24 cm
Thus, height of the given right circular cone = 24 cm
Now, Volume of a Right Circular Cone `=1/3\ pi\ r^2\ h`
Therefore, volume of the given right circular cone
= `1/3xx22/7` × 7 cm × 7 cm × 24 cm
= 22 × 1 cm × 7 cm × 8 cm
= 22 × 56 cm3
= 1232 cm3
Thus, volume of the given right circular cone = 1232 cm3
Now, since 1000 cm3 = 1 Litre
Therefore, 1232 cm3 `= 1/1000 xx1232\ l`
=1.232 litre
Thus, capacity of the given right circular cone = 1.232 litre Answer
(ii) height = 12 cm, slant height 13 cm
Given, height of the right circular cone = 12 cm
And, Slant height of the right circular cone = 13 cm
Therefore, capacity, i.e. volume of the given right circular cone in litres = ?
We know that Volume of a Right Circular Cone `=1/3\ pi\ r^2\ h`
Therefore, it is necessary to calculate radius before calculation of volume of the given right circular cone.
Here, we know that, slant height, radius and height of a right circular cone together form a right angled triangle.
Therefore,
[slant height of a right circular cone (`l`)]2 = (radius)2 + (height)2
Thus, slant height of given right circular cone,
(13 cm)2 = r2 + (12 cm)2
⇒ 169 cm2 = r2 + 144 cm2
⇒ r2 = 169 cm2 – 144 cm2
⇒ r2 = 25 cm2
`=> r = sqrt(25 cm^2)`
⇒ r = 5 cm
Thus, radius of the given right circular cone = 5 cm
Now, volume of the given right circular cone = 1/3 ℿ r2 h
= `1/3xx22/7` × 5 cm × 5 cm × 12 cm
= `22/7` × 25 cm2 × 4 cm
`= 2200 /7\ cm^3` - - - - -(i)
Now, since, 1000 cm3 = 1 Litre
Therefore, 2200/7 cm3 `= 1/1000xx2200/7` Litre
`=22/10xx1/7 = 11/35` Litre
Thus, capacity of the given conical vessel = 11/35 Litres Answer
Or, Capacity of the vessel = 0.314286 Litres Answer
Alternatively from equation (i)
Capacity of the given vessel =2200/7 cm3
= 314.286 cm3
Now, since, 1000 cm3 = 1 Litre
Therefore, 314.286 cm3 `= 1/1000xx314.286` Litre
= 0.314286 Litre
Thus, capacity of the given conical vessel = 0.314286 Litres Answer
Surface Areas And Volumes Class nine Math NCERT Exercise 13.7 Question (3) The height of a cone is 15 cm. If its volume is 1570 cm3, find the radius of the base. (use ℿ = 3.14)
Solution
Given, Height of the cone = 15 cm
And, Volume of the cone = 1570 cm3
And, ℿ = 3.14
Therefore, radius of the base = ?
We know that Volume of a cone = 1/3 ℿ r2 h
Therefore, for the given cone, volume,
1570 cm3 = `1/3` × 3.14 × r2 × 15 cm
⇒ 1570 cm3 = 3.14 × r2 × 5 cm
⇒ 1570 cm3 = r2 × 15.70 cm
`=>r^2 = (1570 cm^3)/(15.70)`
⇒ r2 = 100 cm2
Therefore, `r = sqrt(100 cm^2)`
⇒ r = 10 cm
Thus, radius of the base of the given cone = 10 cm Answer
Surface Areas And Volumes Class nine Math NCERT Exercise 13.7 Question (4) If the volume of a right circular cone of height 9 cm is 48 ℿ cm3, find the diameter of its base.
Solution
Given, Height of the cone = 9 cm
And, Volume of the cone = 48 ℿ cm3
Therefore, Diameter =?
We know that Volume of a cone = 1/3 ℿ r2 h
Therefore, volume of the given cone = `1/3` ℿ r2 × 9 cm
⇒ 48 ℿ cm3 = `1/3` ℿ r2 × 9 cm
⇒ 48 ℿ cm3 = ℿ r2 × 3 cm
`=>r^2 = (48\ pi\ cm^3)/(3cm xx pi)`
⇒ r2 = 16 cm2
Therefore, `r=sqrt(16 cm^2)`
⇒ r = 4 cm
Thus, radius of the given right circular cone = 4 cm
Now, since diameter = 2r
⇒ diameter of the given cone = 2 × 4 cm
⇒ diameter of the given cone = 8 cm
Thus, diameter of the base of the given cone = 8 cm Answer
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