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Answer :

Let ABC be the metallic cone, DECB is the required frustum.

Let the two radii of the frustum be \( DO' \ = \ r_2 \) and \( BO \ = \ r_1 \)

From the ∆ ADO' and ABO,

\( r_2 \ = \ h_1 \) tan 30º \( = \ 10 \ × \ \frac{1}{ \sqrt{3}} \)

\( r_1 \ = \ (h_1 \ + \ h_2 \) tan30º \( = \ 20 \ × \ \frac{1}{ \sqrt{3}} \)

Volume of the frustum DBCE \(= \ \frac{1}{3} \pi h_2 \ × [ (r_1)^2 \ + \ (r_1)(r_2) \ + \ (r_2)^2 ] \)

\(= \ \frac{1}{3} \pi h_2 \ × \ [ \frac{400}{3} \ + \ \frac{200}{3} \ + \ \frac{100}{3}] \ \)

\( = \frac{ \pi \ × \ 10}{3} \ × \ \frac{700}{3} \)

Volume of the wire of length l and diameter D

\(= \ \pi ( \frac{D}{2})^2 \ × \ l \ \)

\( = \ \frac{ \pi D^2}{4} \ × \ l \)

[ \( V \ = \ \pi r2h \) ]

\(\therefore \) Volume of the frustum = Volume of the wire drawn from it

\(\Rightarrow \ \frac{ \pi \ × \ 10}{3} \ × \ \frac{700}{3} \ \)

\( = \ \frac{ \pi D^2}{4} \ × \ l \)

[ \(\because \) D=116]

\(l \ = \ \frac{10 \ × \ 700 \ × \ 4}{3 \ × \ 3 \ × \ 16 \ × \ 16} \ \)

\( = \ 7964.44 \) m

- A drinking glass is in the shape of a frustum of a cone of height 14 cm. The diameters of its two circular ends are 4 cm and 2 cm. Find the capacity of the glass.
- The slant height of a frustum of a cone is 4 cm and the perimeters (circumference) of its circular ends are 18 cm and 6 cm. Find the curved surface area of the frustum.
- A fez, the cap used by the Turks, is shaped like the frustum of a cone (see figure). If its radius on the open side is 10 cm, radius at the upper base is 4 cm and its slant height is 15 cm, find the area of material used for making it.
- A container opened from the top and made up of a metal sheet, is in the form of a frustum of a cone of height 16 cm with radii of its lower and upper ends as 8 cm and 20 cm, respectively. Find the cost of milk which can completely fill the container, at the rate of Rs 20 per litre. Also find the cost of metal sheet used to make the container, if it costs Rs 8 per 100 cm2. (Take \( \pi \ = \ 3.14 \) ).

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