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If the curved surface area of a cone is thrice that of another cone and slant height of the secondcone is thrice that of the first, find the ratio of the area of their base.
- a)81 : 1
- b)9 : 1
- c)3 : 1
- d)27 : 1
Correct answer is option 'A'. Can you explain this answer?
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Given: Curved surface area of cone 1 = 3 × curved surface area of cone 2
Slant height of cone 2 = 3 × slant height of cone 1
To find: Ratio of the area of the base of the cones
Let the radius of the base of cone 1 be r1 and its slant height be l1.
Let the radius of the base of cone 2 be r2 and its slant height be l2.
Formula for the curved surface area of a cone is πrl, where r is the radius of the base and l is the slant height.
From the given information, we have:
πr1l1 = 3πr2l2 => r1l1 = 3r2l2 ------(i)
Also, we know that the formula for the total surface area of a cone is πr(r + l), where r is the radius of the base and l is the slant height.
For cone 1, total surface area = πr1(r1 + l1)
For cone 2, total surface area = πr2(r2 + l2)
We are given that the curved surface area of cone 1 is thrice that of cone 2. Therefore, the total surface area of cone 1 must be thrice that of cone 2 as well. So we have:
πr1(r1 + l1) = 3πr2(r2 + l2) => r1(r1 + l1) = 3r2(r2 + l2) ------(ii)
Dividing equation (ii) by equation (i), we get:
(r1 + l1)/l1 = 3(r2 + l2)/(r2l2)
(r1/l1) + 1 = 3(r2/l2) + 3
(r1/l1) - 3(r2/l2) = 2
Now, using equation (i), we have:
r2 = (r1/l1) * (3l2)
Substituting this in the above equation, we get:
(r1/l1) - 3(r1/l1) * 3 = 2
(r1/l1) = 1/9
Therefore, the ratio of the areas of the bases of the cones is:
πr1^2/πr2^2 = (r1/r2)^2 = (l1/l2)^2 = (1/3)^2 = 1/9
Hence, the answer is option A) 81:1.
Slant height of cone 2 = 3 × slant height of cone 1
To find: Ratio of the area of the base of the cones
Let the radius of the base of cone 1 be r1 and its slant height be l1.
Let the radius of the base of cone 2 be r2 and its slant height be l2.
Formula for the curved surface area of a cone is πrl, where r is the radius of the base and l is the slant height.
From the given information, we have:
πr1l1 = 3πr2l2 => r1l1 = 3r2l2 ------(i)
Also, we know that the formula for the total surface area of a cone is πr(r + l), where r is the radius of the base and l is the slant height.
For cone 1, total surface area = πr1(r1 + l1)
For cone 2, total surface area = πr2(r2 + l2)
We are given that the curved surface area of cone 1 is thrice that of cone 2. Therefore, the total surface area of cone 1 must be thrice that of cone 2 as well. So we have:
πr1(r1 + l1) = 3πr2(r2 + l2) => r1(r1 + l1) = 3r2(r2 + l2) ------(ii)
Dividing equation (ii) by equation (i), we get:
(r1 + l1)/l1 = 3(r2 + l2)/(r2l2)
(r1/l1) + 1 = 3(r2/l2) + 3
(r1/l1) - 3(r2/l2) = 2
Now, using equation (i), we have:
r2 = (r1/l1) * (3l2)
Substituting this in the above equation, we get:
(r1/l1) - 3(r1/l1) * 3 = 2
(r1/l1) = 1/9
Therefore, the ratio of the areas of the bases of the cones is:
πr1^2/πr2^2 = (r1/r2)^2 = (l1/l2)^2 = (1/3)^2 = 1/9
Hence, the answer is option A) 81:1.
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If the curved surface area of a cone is thrice that of another cone and slant height of the secondcone is thrice that of the first, find the ratio of the area of their base.a)81 : 1b)9 : 1c)3 : 1d)27 : 1Correct answer is option 'A'. Can you explain this answer?
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