Question : The radius of the base of a solid cone is 9 cm, and its height is 21 cm. It was cut into three parts by two cuts, which were parallel to its base. The cuts are at a height of 7 cm and 14 cm from the base, respectively. What is the ratio of the curved surface area of the top, middle, and bottom parts, respectively?
Option 1: $1 : 4: 8$
Option 2: $1 : 3: 5$
Option 3: $1 : 3: 9$
Option 4: $1: 6: 12$
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Correct Answer: $1 : 3: 5$
Solution : Given: Radius = 9 cm, height = 21 cm Two cuts at h = 7 cm and 14 cm Height will be divided into 3 parts and the height for three cones will be 7 cm, 14 cm, and 21 cm. $\therefore$ Radius will be 3 cm, 6 cm, and 9 cm. New slant height $l_1=\sqrt{7^2+3^2}=7.6$ cm $l_2=\sqrt{14^2+6^2}=15.2$ cm $l_3=\sqrt{21^2+9^2}=22.8$ cm The ratio of the curved surface of the three cones is: $\pi r_1 l_1:\pi r_2 l_2:\pi r_3 l_3$ $=7.6\times7:15.2\times14:22.8\times21$ $=1:4:9$ So, the ratio of the curved surface area of the top, middle, and bottom parts, respectively = $1:(4-1):(9-4)$ = $1:3:5$ Hence, the correct answer is $1: 3: 5$.
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