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A right circular cone is sliced into a smaller cone and a frustum of a cone by a plane perpendicular to its axis. The volume of the smaller cone and the frustum of the cone are in the ratio 64 ∶ 61. Then their curved surface areas are in the ratio
- a)4 ∶ 1
- b)16 ∶ 9
- c)64 ∶ 61
- d)81 ∶ 64
Correct answer is option 'B'. Can you explain this answer?
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Verified Answer
A right circular cone is sliced into a smaller cone and a frustum of a...
For same vertical angle,
Radius of small cone/Radius of large cone = Height of small cone/Height of large cone Slant height of small cone/Slant height of the large cone = k
Volume of small cone∶ Volume of frustum = 64 ∶ 61
∴ Volume small cone∶ Volume large cone = 64 ∶ (64 + 61) = 64 ∶ 125
Let the radii, height and slant heights of the small and large cones be r, h, l and R, H, L
Volume of small cone/Volume of large cone = 64/125
(1/3 π r2 h) / (1/3 π R2 H) = 64/125
(r/R)2 × (h/H) = 64/125
k2 × k = 64/125
k3 = 64/125
∴ k = 4/5
(r/R)2 × (h/H) = 64/125
k2 × k = 64/125
k3 = 64/125
∴ k = 4/5
Curved surface area of small cone/Curved surface area of large cone = (π r l) / (π R L)
⇒ (r/R) × (l/L)
⇒ k × k = k2 = 16/25
⇒ (r/R) × (l/L)
⇒ k × k = k2 = 16/25
Curved surface area of small cone/Curved surface area of frustum = 16/ (25 – 16) = 16/9
∴ Required ratio 16 ∶ 9
∴ Required ratio 16 ∶ 9
Most Upvoted Answer
A right circular cone is sliced into a smaller cone and a frustum of a...
Explanation:
Given: Volume ratio of the smaller cone to the frustum of the cone = 64 : 61
Volume Ratio:
- Let the volumes of the smaller cone and the frustum of the cone be V1 and V2 respectively.
- According to the given ratio, V1 : V2 = 64 : 61
Volume Formula:
- The volume of a cone is given by V = (1/3) * π * r^2 * h, where r is the radius and h is the height of the cone.
Relationship:
- Let the smaller cone have radius r1 and height h1, and the frustum of the cone have radii r2 and R, and height h2.
- Since the smaller cone is obtained by slicing the larger cone, the heights of the two cones are proportional. Hence, h1 : h2 = V1^(1/3) : V2^(1/3)
Curved Surface Area Ratio:
- The curved surface area of a cone is given by A = π * r * l, where l is the slant height of the cone.
Using Volume Formula:
- From the volume formula, we can express the radii of the two cones in terms of their heights: r1 = (3V1 / πh1)^(1/2) and r2 = (3V2 / πh2)^(1/2)
Using Curved Surface Area Formula:
- Substituting the values of r1, r2, h1, and h2 into the formula for the curved surface area, we get the ratio of the curved surface areas of the smaller cone to the frustum as (64/61)^(2/3) = 16/9.
Therefore, the curved surface areas of the smaller cone and the frustum of the cone are in the ratio 16 : 9. Hence, the correct answer is option 'B'.
Given: Volume ratio of the smaller cone to the frustum of the cone = 64 : 61
Volume Ratio:
- Let the volumes of the smaller cone and the frustum of the cone be V1 and V2 respectively.
- According to the given ratio, V1 : V2 = 64 : 61
Volume Formula:
- The volume of a cone is given by V = (1/3) * π * r^2 * h, where r is the radius and h is the height of the cone.
Relationship:
- Let the smaller cone have radius r1 and height h1, and the frustum of the cone have radii r2 and R, and height h2.
- Since the smaller cone is obtained by slicing the larger cone, the heights of the two cones are proportional. Hence, h1 : h2 = V1^(1/3) : V2^(1/3)
Curved Surface Area Ratio:
- The curved surface area of a cone is given by A = π * r * l, where l is the slant height of the cone.
Using Volume Formula:
- From the volume formula, we can express the radii of the two cones in terms of their heights: r1 = (3V1 / πh1)^(1/2) and r2 = (3V2 / πh2)^(1/2)
Using Curved Surface Area Formula:
- Substituting the values of r1, r2, h1, and h2 into the formula for the curved surface area, we get the ratio of the curved surface areas of the smaller cone to the frustum as (64/61)^(2/3) = 16/9.
Therefore, the curved surface areas of the smaller cone and the frustum of the cone are in the ratio 16 : 9. Hence, the correct answer is option 'B'.
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A right circular cone is sliced into a smaller cone and a frustum of a cone by a plane perpendicular to its axis. The volume of the smaller cone and the frustum of the cone are in the ratio 64 61. Then their curved surface areas are in the ratioa)4 1b)16 9c)64 61d)81 64Correct answer is option 'B'. Can you explain this answer?
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