Volume of Cone

Exercise 13.7

Question 1: Find the volume of the right circular cone with

(i) radius 6 cm, height 7 cm

Answer: Given; r = 6 cm, h = 7 cm
Volume of cone `= 1/3\ π r^2\h`
`= (1/3) xx (22/7) xx 6^2\ xx 7 = 264` cubic cm

(ii) radius 3.5 cm, height 12 cm

Answer: Given; r = 3.5 cm, h = 12 cm
Volume of cone `= 1/3\ π r^2\ h`
`= (1/3) xx (22/7) xx (3.5)^2\ xx 12 = 154` cubic cm

Question 2: Find the capacity in litres of a conical vessel with

(i) radius 7 cm, slant height 25 cm

Answer: Given; `r = 7` cm, `l = 25` cm
Here; `h^2= l^2 - r^2`
`= 25^2 - 7^2`
`= 625 – 49 = 576`
Or, `h = 24` cm

Volume of cone `= 1/3 \π r^2\ h`
`= (1/3) xx (22/7) xx 7^2\ xx 24 = 1232` cubic cm
= 1.232 litre

(ii) height 12 cm, slant height 13 cm

Answer: Given; `h = 12` cm, `l = 13 `cm
Here, `r^2 = l^2 - h^2`
`= 13^2 - 12^2`
`= 169 – 144 = 25`
Or, `r = 5` cm

Volume of cone `= 1/3\ π r^2\ h`
`= (1/3) xx (3.14) xx 5^2\ xx 12 = 314` cubic cm
= 0.314 litre


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)

Answer: Given; volume = 1570 cubic cm, h = 15 cm, r = ?
Volume of cone `= 1/3\ π r^2\h`
Or, `1570 = (1/3) xx (3.14) xx r^2\ xx 15`
Or, `r^2 = (1570 xx 3)/(3.14 xx 15) = 100`
Or, `r = 10` cm

Question 4: If the volume of a right circular cone of height 9 cm is 48 π cm3, find the diameter of its base.

Answer: Given; volume = 48 π cubic cm, h = 9 cm, r = ?
Volume of cone `= 1/3\ π r^2\ h`
Or, `48 π = 1/3 \π r^2\ xx 9`
Or, `r^2 = 48/3 = 16`
Or, `r = 4` cm


Question 5: A conical pit of top diameter 3.5 m is 12 m deep. What is its capacity in kilolitres?

Answer: Given; r = 3.5 m, h = 12 m
Volume of cone `= 1/3 \π r^2\ h`
`= (1/3) xx (22/7) xx (3.5)^2\ xx 12 = 154` cubic cm
= 154 kilo litre

Question 6: The volume of a right circular cone is 9856 cm3. If the diameter of the base is 28 cm, find
(i) height of the cone
(ii) slant height of the cone
(iii) curved surface area of the cone

Answer: Given; volume = 9856 cubic cm, d = 28 cm so, r = 14 cm
Volume of cone `= 1/3\ π r^2\ h`
Or, `9856 = (1/3) xx (22/7) xx 14^2\ xx h`
Or, `h = (9856 xx 3 xx 7)/(22 xx 196) = 48` cm

Slant Height:
Here: `l^2 = h^2 + r^2`
`= 48^2 + 14^2 = 2500`
Or, `l = 50` cm

Curved surface area of cone `= π rl`
`= (22/7) xx 14 xx 50 = 2200` sq cm

Question 7: A right triangle ABC with sides 5 cm, 12 cm and 13 cm is revolved about the side 12 cm. Find the volume of the solid so obtained.

Answer: Given; h = 12 cm, r = 5 cm
Volume of cone `= 1/3 \π r^2\ h`
`= (1/3) π xx 5^2\ xx 12 = 100 π` cubic cm

Question 8: If the triangle ABC in the Question 7 above is revolved about the side 5 cm, then find the volume of the solid so obtained. Find also the ratio of the volumes of the two solids obtained in Questions 7 and 8.

Answer: Given; h = 5 cm, r = 12 cm
Volume of cone `= 1/3 \π r^2\ h`
`= (1/3) π xx 12^2\ xx 5 = 240 π` cubic cm
Ratio `= 100/240 = 5 : 12`

Question 9: A heap of wheat is in the form of a cone whose diameter is 10.5 m and height is 3 m. Find its volume. The heap is to be covered by canvas to protect it from rain. Find the area of the canvas required.

Answer: Given; r = 5.25 m, h = 3 m
Volume of cone `= 1/3\ π r^2\h`
`= (1/3) xx (22/7) xx (5.25)^2\ xx 3`
= 50.625 cubic m



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