Right Circular Cone
Class 10th Mathematics West Bengal Board Solution
- I have made a closed right circular cone whose length of the base radius is 15cm and…
- Let us determine the volume of the cone when, (i) base area is 1.54 sq. m and height is…
- Amina has drawn a right-angled triangle whose lengths of two sides adjacent to right…
- If the height and slant height of a cone are6cm.and 10cm. respectively, then let us…
- If the volume of a right circular cone is 100 π cm^3 and height is 12cm, then let us…
- 77sq.m. tripal is required to make a right circular conical tent. If the slant height…
- The base area of a right circular cone is 21m. and height is 14m. Let us calculate the…
- The length of the base diameter of a wooden toy of the conical shape is 10cm. The…
- The quantity of iron sheet to make boya of right circular conical shape is 75 3/7 m^2 .…
- In a right circular conical tent 11 persons can stay. For each person 4m^2 space in…
- The external diameter of a conical-coronet made off thermocol is 21cm. in length. To…
- A heap of wheat is in the shape of a right circular cone, its base diameteris 9m. and…
- Q13A1 If the slant height of a right circular cone is 15 cm. and the length of the base…
- Q13A2 If the ratio of the volumes of two right circular cones is 1:4 and the ratio of…
- Q13A3 Keeping the radius of a right circular cone same, if the height of it is increased…
- Q13A4 If each of radius of a cone is increased by twice of its length, then the volume of…
- Q13A5 If the length of the radius of a cone is r/2 unit and slant height of it is 2l unit,…
- Let us write whether the following statements are true or false: (i) If the length of…
- Let us fill in the blanks: (i) AC is the hypotenuse of a right-angle triangle ABC,…
- The height of a right circular cone is 12cm. and its volume is 100 π cm^3 . Let us…
- The curved surface area of a right circular cone is √5 times of its base area. Let us…
- If the volume of a right circular cone is V cubic unit, base area A sq. unit and…
- The numerical values of the volume and the lateral surface area of a right circular…
- The ratio of the lengths of the base radii of a right circular cylinder and a right…
Let Us Work Out 16
Question 1.I have made a closed right circular cone whose length of the base radius is 15cm and slant height is 24cm. Let us calculate the curved surface area and total surface area of that cone.
Answer:
Base radius of the cone, r = 15 cm
Slant height of the cone, l = 24 cm
∴ Height of the cone, h = √(l2 – r2) = √(242 – 152) = √(576 – 225) = √351 = 18.73 cm
∴ Curved surface area of the cone,
= πrl cm2
= 
× 15 × 24 cm2
= 1131.43 cm2
∴ Total surface area,
= πr2 + πrl cm2
= 22/7 × 152 + 1131.43 cm2
= 707.14 + 1131.43 cm2
= 1838.57 cm2
Question 2.
Let us determine the volume of the cone when, (i) base area is 1.54 sq. m and height is 2.4m (ii) the length of base diameter is 21m and slant height is 17.5m.
Answer:
i) Base area, πr2 = 1.54 sq.m.
height, h = 2.4 m
∴ volume of the cone,
= πr2 × h/3 m3
= 1.54 × 2.4/3 m3
= 1.232 m3
ii) base diameter = 21 m
∴ base radius, r = 21/2 = 10.5 cm
Slant height, l = 17.5 m
∴ height, h = √(l2 – r2) = √(17.52 – 10.52) = √(306.25 – 110.25) = √196 = 14 m
∴ volume of the cone,
= πr2h/3
= 22/7 × 10.52 × 14/3
= 1617 m3
Question 3.
Amina has drawn a right-angled triangle whose lengths of two sides adjacent to right angle are 15cm and 20 cm. Let us determine the curved surface area, total surface area and volume of the solid which is formed by taking the side of length 15cm. which is formed by completely revolving the triangle once around the side of the triangle with the length of 15cm, having been taken as an axis.
Answer:
The solid formed by taking the side of length 15 cm and completely revolving the triangle once is a right circular cone.
According to problem,
Height of the cone, h = 15 cm
Length of base radius, r = 20 cm
∴ slant height, l = √(h2 + r2)√(152 + 202) = √(225 + 400) = 25 cm
∴ curved surface area,
= πrl
= 22/7 × 20 × 25
= 1571.43 sq. cm.
∴ Total surface area,
= πr2 + πrl
= 22/7 × 202 + 1571.43
= 1257.14 + 1571.43
= 2828.57 sq cm.
∴ volume,
= πr2h/3
= 22/7 × 202 × 15/3
= 6285.71 cubic cm.
Question 4.
If the height and slant height of a cone are6cm.and 10cm. respectively, then let us determine total surface area and volume of the cone.
Answer:
Height, h = 6 cm
Slant height, l = 10 cm
∴ radius, r = √(l2 – h2) = √(102 – 62) = √(100 – 36) = 8 cm
∴ Total surface area,
= πr2 + πrl
= 22/7 × 82 + 22/7 × 8 × 10
= 201.14 + 251.43
= 452.57 sq. cm.
∴ volume,
= πr2h/3
= 22/7 × 82 × 6/3
= 402.28 cubic cm
Question 5.
If the volume of a right circular cone is 100 π cm3 and height is 12cm, then let us write by calculating, the slant height of the cone.
Answer:
Let, base radius = r cm and slant height = l cm
Height = 12 cm
According to problem,
⇒ π × r2 ×h/3 = 100π
⇒ 4r2 = 100
⇒ r = 5
Base radius = 5 cm
∴ Slant height = √(h2 + r2) = √(122 + 52) = √(144 + 25) = 13 cm
Question 6.
77sq.m. tripal is required to make a right circular conical tent. If the slant height is 7m. Then let us write by calculating, the base area of the tent.
Answer:
Structure of a tent is like a right circular cone.
∴ tripal required to make tent = curved surface area of the cone.
Let, base radius of the tent = r m
Slant height = 7 m
According to problem,
⇒ π × r × 7 = 77
⇒ 22/7 × r = 11
⇒ r = 7/2 = 3.5
∴ base radius of the tent = 3.5 m
∴ Base area of the tent,
= πr2
= 22/7 × 3.52
= 38.5 sq. m
Question 7.
The base area of a right circular cone is 21m. and height is 14m. Let us calculate the expenditure to colour the curved surface at the rate of ₹1.50 per sq.m.
Answer:
Let, base radius of the cone = r m
According to problem,
⇒ πr2 = 21
⇒ r2 = 21 ×7/22
⇒ r = 2.58
∴ radius of the base = 2.58 m
Height of the cone = 14 m
∴ slant height of the cone,
= √(2.582 + 142)
= √(6.66 + 196)
= 14.24 cm
∴ curved surface area of the cone,
= πrl
= 22/7 × 2.58 × 14.24
= 115.47 sq. m
∴ Expenditure to colour the curved surface area,
= 115.47 × 1.50
= Rs. 173.2
Question 8.
The length of the base diameter of a wooden toy of the conical shape is 10cm. The expenditure for polishing the whole surfaces of the toy at the rate of ₹2.10 per m2 is ₹429. Let us calculate the height of the toy. Let us also determine the quantity of wood which is required to make the toy.
Answer:
Let, whole surface area of the toy = x m2
According to problem,
⇒ x × 2.1 = 429
⇒ x = 429/2.1
⇒ x = 204.286
Base diameter of the toy = 10 cm
∴ base radius of the toy, r = 10/2 = 5 cm
Let, slant height = l cm
According to problem,
⇒ πr2 + πrl = 204.286
⇒ 22/7(52 + 5l) = 204.286
⇒ 5l + 25 = 65
⇒ 5l = 40
⇒ l = 8
∴ slant height = 8 cm
∴ height = √(l2 – r2)= √(82 – 52) = √(64 – 25) = 6.24 cm
∴ Volume of the toy,
= πr2h/3
= 22/7 × 52 × 6.24/3
= 163.43 m3
∴ Quantity of wood required to make the toy = 163.43 m3
Question 9.
The quantity of iron sheet to make boya of right circular conical shape is 
m2. If the slant height of it is 5m, then let us write, by calculating, the volume of air in the boya and its height. Let us determine the expenditure to colour the whole surface of the boya at the rate of ₹2.80 per m2. [The width of the iron-sheet not to be considered while calculating.]
Answer:
Let, base radius = r m
Slant height, l = 5 m
According to problem,
⇒ r2 + 5r = 24
⇒ r2 + 8r – 3r – 24 = 0
⇒r(r + 8) – 3(r + 8) = 0
⇒ (r + 8)(r – 3) = 0
⇒ r = -8 or 3
∴ base radius = 3 m
∴ height of the boya, h = √(52 – 32)
= √(25 – 9) = 4 m
∴ volume of air in the boya,
= πr2h/3
= 22/7 × 32 × 4/3
⇒ 37.71 m3
Question 10.
In a right circular conical tent 11 persons can stay. For each person 4m2 space in the base and 20m3 air are necessary. Let us determine the height of the tent put up exactly for 11 persons.
Answer:
Let, base radius of the tent = r m
Height of the tent = h m
Space in the base in the tent = πr2 m2
According to problem,
⇒ πr2 = 11 × 4
⇒ πr2 = 44 ……….. (1)
Volume of air in the tent = ⇒ πr2h/3
According to problem,
⇒ πr2h/3 = 11 × 20
⇒ 44 × h = 660 [putting the value from (1)]
⇒ h = 15
∴ height of the tent = 15 m
Question 11.
The external diameter of a conical-coronet made off thermocol is 21cm. in length. To wrap up the outer surface of the coronet with foil, the expenditure will be ₹57.75 at the rate of 10p. per cm2. Let us write by calculating, the height and slant height of the coronet.
Answer:
External diameter of coronet = 21 cm
∴ external radius of coronet, r = 21/2 = 10.5 cm
Let, slant height = l cm
Height = h cm
According to problem,
⇒πrl = 57.75/0.1 [10p. = Rs. 0.1]
⇒ 22/7 × 10.5 × l = 577.5
⇒ l = 577.5 × 7/(22 × 10.5)
⇒ l = 17.5
∴ slant height = 17.5 cm
∴ height = √(17.52 – 10.52) = √(306.25 – 110.25) = 14 cm
Question 12.
A heap of wheat is in the shape of a right circular cone, its base diameteris 9m. and height is 3.5m. Let us determine the total volume of wheat. Let us calculate the minimum quantity of plastic sheet to be required to cover up this heap of wheat [suppose π =3.14, √130 = 11.1]
Answer:
Base diameter = 9 m
∴ base radius = 9/2 = 4.5 m
Height, h = 3.5 m
∴ volume of wheat,
= πr2h/3
= 3.14 × 4.52× 5/3
= 105.975 m3
Slant height,
l = √(4.52 + 3.52) = √(20.25 + 12.25) = 5.7 m
∴ Plastic sheet required to cover the wheat,
= πrl
= 3.14 × 4.5 × 5.7
= 80.541 m2
Question 13.
If the slant height of a right circular cone is 15 cm. and the length of the base diameter is 16cm, then the lateral surface area of the cone is
A. 60π
B. 68π cm2
C. 120π cm2
D. 130π cm2
Answer:
Slant height, l = 15 cm
Base diameter = 16 cm
∴ base radius, r = 16/2 = 8 cm
∴ lateral surface area of cone,
= πrl
= π × 8 × 15
= 120π cm2
Question 14.
If the ratio of the volumes of two right circular cones is 1:4 and the ratio of their radii of their bases is 4:5, then the ratio of the heights is
A. 1:5
B. 5:4
C. 25:16
D. 25:64
Answer:
Volume of the first cone = v and second cone = 4v
First cone’s base radius = 4r and second cone’s base radius = 5r
Let, first cone’s height = x and second cone’s height = y
According to the problem,
⇒ x/y = 25/64
⇒x∶ y = 25 ∶ 64
Question 15.
Keeping the radius of a right circular cone same, if the height of it is increased twice, the volume of it will be
A. 100%
B. 200%
C. 300%
D. 400%
Answer:
In 1st case,
Base radius = r
Height = h
∴ Volume, x = πr2h/3
In 2nd case,
Base radius = r
Height = 2h
∴ volume, y = πr2×2h/3 = 2πr2h/3
∴ increase in volume,
= 100%
∴ volume will be 200%
Question 16.
If each of radius of a cone is increased by twice of its length, then the volume of it will be
A. 3 times
B. 4 times
C. 6 times
D. 8 times of the previous one.
Answer:
In 1st case,
Base radius = r
Height = h
∴ Volume, x = πr2h/3
In 2nd case,
Base radius = 2r
Height = h
∴ volume, y = π(2r)2×h/3 = 4πr2h/3
⇒ y/x = 4
⇒ y = 4x
∴ The volume will be 4 times.
Question 17.
If the length of the radius of a cone is 
unit and slant height of it is 2l unit, then the total surface area is
A. 
sq.unit
B. 
sq.unit
C. 
sq. unit
D. 
sq.unit
Answer:
Radius = r/2 unit
Slant height = 2l unit
∴ Total surface area,
Question 18.
Let us write whether the following statements are true or false:
(i) If the length of the base radius of a right circular cone is decreased by half and its height is increased by twice of it then the volume remains the same.
(ii) The height, radius and slant height of a right circular cone are always the three sides of a right-angles triangles.
Answer:
(i) false
In 1st case,
Base radius = r
Height = h
∴ Volume, x = πr2h/3
In 2nd case,
Base radius = r/2
Height = 2h
∴ volume, y = π(r/2)2×2h/3 = πr2h/6
∴ x ≠ y
∴ volume will not remain same
(ii) True
The height is equivalent to the perpendicular of a right-angle triangle.
The radius is equivalent to the base and the slant height is equivalent to the hypotenuse.
Question 19.
Let us fill in the blanks:
(i) AC is the hypotenuse of a right-angle triangle ABC, the radius of the right circular cone formed by revolving the triangle once around the side AB as the axis is___
(ii) If the volume of a right circular cone is V cubic unit and the base area is A sq. unit, then its height is____
(iii) The lengths of the base radii and the heights of a right circular cylinder and a right circular cone are equal. The ratio of their volume is_____.
Answer:
(i) BC
AC is equivalent to the slant height.
AB is equivalent to height.
BC is equivalent to radius.
(ii) We know,
Volume of a right circular cone = base area × height
⇒ V = A × height
⇒height = V/A
(iii) In 1st case,
Base radius = r
Height = h
∴ Volume, x = πr2h/3
In 2nd case,
Base radius = r
Height = h
∴ volume, y = πr2h/3
∴ x∶ y = (πr2h/3) ∶ (πr2h/3) = 1 ∶ 1
Question 20.
The height of a right circular cone is 12cm. and its volume is 100 π cm3. Let us write the length of the radius of the cone.
Answer:
Let, radius = r cm
Height = 12 cm
According to problem,
⇒π × r2 × 12 = 100π
⇒ 12r2 = 100
⇒ r2 = 25/3
⇒ r = 2.89
∴ Length of radius of the cone = 2.89 cm
Question 21.
The curved surface area of a right circular cone is √5 times of its base area. Let us write the ratio of the height and the length of a right circular cone are always the three sides of a right-angled triangle.
Answer:
Given: Curved Surface Area of a right circular cone = √5 Base Area of right circular cone
Curved Surface Area of a right circular cone = πrl
Base area of right circular cone, A = πr2
……………….eq(i)
Now from the cone we know that,
……………..eq(ii)
Where, h is the height of cone
Now from (i) and (ii), we get that
Now, h cannot be negative, hence such cone is not possible.
Question 22.
If the volume of a right circular cone is V cubic unit, base area A sq. unit and height is H unit, then let us write the value of 
.
Answer:
We know,
Volume of a right circular cone = Base area × Height
⇒ V = A × H
⇒ AH/V = 1
Question 23.
The numerical values of the volume and the lateral surface area of a right circular cone are equal. If the height and the radius of the cone are h unit and r unit respectively, then let us write the value of 
.
Answer:
Height = h unit
Radius = r unit
∴ slant height, l = √(h2 + r2)
According to problem,
⇒ r2 + h2 = r2h2/9
Question 24.
The ratio of the lengths of the base radii of a right circular cylinder and a right circular cone is 3:4 and the ratio of their heights is 2:3; let us write the ratio of the volumes of the cylinder and the cone.
Answer:
Let, base radius of the cone = 4r
Base radius of the cylinder = 3r
Let, the height of the cone = 3h
The height of the cylinder = 2h
∴ volume of the cone,
= 1/3 × π × (4r)2 × 3h
= 16πr2h
∴ volume of the cylinder,
= π × (3r)2 × 2h
⇒ 18πr2h
∴ ratio of the volumes of the cylinder and the cone,
= (16πr2h) ∶ (18πr2h)
= 8 ∶ 9