A circular cone is cut by a plane parallel to the base and the conical portion is removed. If the curved surface area of the frustum is 15 16 of the curved surface area of the whole cone, prove that the height of the frustum is 3 4 of the height of the whole cone.

Given: Let ABC is the cone of height H with radius R and slant L Let AXY is the cone of height h with radius r and slant l Suppose it is cut by a plane parallel to the base. AO AP = OC PY = AC AY H h = R r = L l ...(i) From the given, the curved surface area of the frustum is 15 16 of the curved surface area of the whole cone Curved surface area of the smaller cone = Curved surface area of cone - Curved surface area of the frustum Curved surface area of the smaller cone = Curved surface area of the whole cone - 15 16 Curved surface area of the whole cone. Curved surface area of the smaller cone = 1 - 15 16 Curved surface area of the whole cone. Curved surface area of the smaller cone = 1 16 Curved surface area of the whole cone. ∴ Curved surface area of the smaller cone Curved surface area of the whole cone = 1 16 ⇒ πrl πRL = 1 16 Using (i) r × l R × L = h 2 H 2 = 1 16 h H = 1 4 ∴ h = H 4 height of the frustum height of the whole cone = H - h H = H - H 4 H = 3 H 4 H = 3 H 4 × 1 H height of the frustum height of the whole cone = 3 4 Hence, proved the height of the frustum is 3 4 of the height of the whole cone.

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The perimeters of circular ends of a solid frustum are $88 cm & 66 cm$ and its slant height is $21 cm$ , find the total surface area of the frustum (in decimal form).

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Important Questions on Mensuration

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10th Manipur Board

IMPORTANT

Mathematics for Class 10>Chapter 12 - Mensuration>EXERCISE 12.6>Q 7

A container is in the form of a frustum of height

12 cm

with radii of its upper and lower ends as

17 cm and 8 cm

respectively. Find the cost of milk the container can hold at the rate of

₹ 20

per litre. Also find the curved surface area of the container (take π=3.14 ).

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10th Manipur Board

IMPORTANT

Mathematics for Class 10>Chapter 12 - Mensuration>EXERCISE 12.6>Q 8

A cone of height

h cm

, is divided into two parts by a plane through the midpoint of the axis of the cone and parallel to the base. Find the ratio of the volume of the conical part to that of the frustum.

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10th Manipur Board

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Mathematics for Class 10>Chapter 12 - Mensuration>EXERCISE 12.6>Q 9

A cone is divided by plane parallel to its base into a smaller cone of volume

v_{1}

and a frustum of volume

v_{2}

. If

v_{1} : v_{2} = 8 : 19

, find the ratio of the radius of the smaller cone to that of the given cone.

HARD

10th Manipur Board

IMPORTANT

Mathematics for Class 10>Chapter 12 - Mensuration>EXERCISE 12.6>Q 10

A circular cone has a base of radius

10 cm

and height

25 cm

. The area of the cross-section of the cone by a plane parallel to its base is

154 {cm}^{2}

. Find the distance of the plane from the base of the cone.

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10th Manipur Board

IMPORTANT

Mathematics for Class 10>Chapter 12 - Mensuration>EXERCISE 12.6>Q 11

A circular cone is cut by a plane parallel to the base and the conical portion is removed. If the curved surface area of the frustum is

\frac{15}{16}

of the curved surface area of the whole cone, prove that the height of the frustum is

\frac{3}{4}

of the height of the whole cone.

HARD

10th Manipur Board

IMPORTANT

Mathematics for Class 10>Chapter 12 - Mensuration>EXERCISE 12.6>Q 12

From a cone of height

24 cm

, a frustum is cut off by a plane parallel to the base of the cone. If the volume of the frustum is

\frac{19}{27}

of the volume of the cone, find the height of frustum.

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10th Manipur Board

IMPORTANT

Mathematics for Class 10>Chapter 12 - Mensuration>EXERCISE 12.6>Q 13

A cone is cut into three parts by planes through the points of trisection of its altitude and parallel to the base. Prove that the volumes of the parts are in the ratio

1 : 7 : 19

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10th Manipur Board

IMPORTANT

Mathematics for Class 10>Chapter 12 - Mensuration>EXERCISE 12.6>Q 14

A bucket is in the form of a frustum with a capacity of

45584 cubic cm

. If the radii of the top and bottom of the bucket are

28 cm & 7 cm

respectively, find its height and surface area.

The perimeters of circular ends of a solid frustum are 88 cm & 66 cm and its slant height is 21 cm, find the total surface area of the frustum (in decimal form).

Important Questions on Mensuration

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The perimeters of circular ends of a solid frustum are $88 cm & 66 cm$ and its slant height is $21 cm$ , find the total surface area of the frustum (in decimal form).