The height of a cone is 30 c m . A small cone is cut off at the top by a plane parallel to its base. If its volume be 1 27 of the volume of the given cone, at what height above the base is the section cut?

Height of a cone = 30 c m A small cone is cut off from the top of the cone Therefore, Volume of small cone = 1 27 of the volume of the given cone. Volume of given cone = 1 3 π R 2 H and volume of small cone = 1 3 π r 2 h So, 1 3 π r 2 h = 1 27 × 1 3 π R 2 H r 2 R 2 h = 1 27 × 30 r 2 R 2 h = 10 9 . . . . . . ( i ) But r R = h H = h 30 ⇒ r R = h 30 . . . . . ( i i ) From i and i i h 30 2 h = 10 9 ⇒ h 2 × h 900 = 10 9 h 3 = 10 × 900 9 = 1000 = 10 c m A line parallel to base at a distance of 30 - 10 = 20 c m

A bucket in the form of a frustum of a cone of height 30 cm with radii of its lower and upper ends as 10 cm and 20 cm , respectively. Find the capacity of the bucket. Also find the cost of milk which can completely fill the bucket at the rate of ₹ 40 per litre. Use π = 22 7

Given that, a bucket in the form of a frustum of a cone of height 30 cm with radii of its lower and upper ends as 10 cm and 20 cm , respectively. Capacity of the bucket can be obtained by finding the volume of the frustum of cone. We know that, volume of the frustum of cone = 1 3 πh r 1 2 + r 2 2 + r 1 r 2 Where, h is the height the frustum of the cone, r 1 , r 2 are the radii of both ends of the frustum. Using the formula mentioned above, Capacity = 1 3 × 22 7 × 30 20 2 + 10 2 + 20 × 10 = 22 7 × 10 400 + 100 + 200 = 22 7 × 10 × 700 = 22 × 10 × 100 = 22000 cm 3 = 22 litre [ ∵ 1 litre = 1000 cm 3 ] Cost of 22 litre milk = Cost of 1 litre milk × 22 = ₹ 40 × 22 = ₹ 880 Hence, the capacity of the bucket is 22 litre and the cost of milk is ₹ 880 .

E M B I B E

CBSE Syllabus Standard Mathematics for Class X>Chapter 14 - Surface Areas and Volumes>Exercise 14.3>Q 6

MEDIUM

10th CBSE

IMPORTANT

Earn 100

A conical vessel, with base radius $5 cm$ and height $24 cm,$ is full of water. This water is emptied into a cylindrical vessel of base radius $10 cm .$ Find the height to which the water will rise in the cylindrical vessel.

Important Questions on Surface Areas and Volumes

HARD

10th CBSE

IMPORTANT

CBSE Syllabus Standard Mathematics for Class X>Chapter 14 - Surface Areas and Volumes>Exercise 14.3>Q 9

A well of diameter

3 m

is dug

14 m

deep. The earth taken out of it has been spread evenly all around it in the shape of a circular ring of width

4 m

to form an embankment. Find the height of the embankment.

MEDIUM

10th CBSE

IMPORTANT

CBSE Syllabus Standard Mathematics for Class X>Chapter 14 - Surface Areas and Volumes>Exercise 14.3>Q 11

There is a water to the height of

14 cm

in a cylindrical glass jar of radius

8 cm

. Inside the water there is a sphere of diameter

12 cm

completely immersed. By what height will the water go down when the sphere is removed ?

HARD

10th CBSE

IMPORTANT

CBSE Syllabus Standard Mathematics for Class X>Chapter 14 - Surface Areas and Volumes>Exercise 14.3>Q 16

Find the number of metallic circular disc with $1.5 c m$ base diameter and of height $0.2 c m$ to be melted to form a right circular cylinder of height $10 c m$ and diameter $4.5 c m$ .

MEDIUM

10th CBSE

IMPORTANT

CBSE Syllabus Standard Mathematics for Class X>Chapter 14 - Surface Areas and Volumes>Exercise 14.3>Q 17

150

spherical marbles, each of diameter

1.4 cm

are dropped in a cylindrical vessel of diameter

7 cm

containing some water, which are completely immersed in water. Find the rise in the level of water in the vessel.

MEDIUM

10th CBSE

IMPORTANT

CBSE Syllabus Standard Mathematics for Class X>Chapter 14 - Surface Areas and Volumes>Exercise 14.4>Q 7

A cone is of height $24 c m$ and the base radius $9 c m .$ Find the radius of the circular section cut from the cone by a plane parallel to the base and $8 c m$ from it.

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