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

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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.

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Important Questions on Mensuration

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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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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.

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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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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