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Two right circular cones have equal radii. If their slant heights are in the ratio 4 : 3, then their respective surface areas are in the ratio
- a)3 : 4
- b)16 : 9
- c)6 : 8
- d)4 : 3
Correct answer is option 'D'. Can you explain this answer?
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Verified Answer
Two right circular cones have equal radii. If their slant heights are ...
Let l1 and l2 be the slant heights of two cones respectively.
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Two right circular cones have equal radii. If their slant heights are ...
{NOTE--Π=pie}
it is given that the radii of both the circular cone are equal so let the radii=r,
or, we can say that the ratio of their radii=1:1--(1).
ratio of their heights=4:3,--(2).
we have to find--> the ratio of their surface area=?
or, Πrl1/Πrl2=?
( from (1).and (2).)
Πrl1/Πrl2=l1/l2( here I denotes height of respective cone)
or,
the ratio of their surface area=4:3
it is given that the radii of both the circular cone are equal so let the radii=r,
or, we can say that the ratio of their radii=1:1--(1).
ratio of their heights=4:3,--(2).
we have to find--> the ratio of their surface area=?
or, Πrl1/Πrl2=?
( from (1).and (2).)
Πrl1/Πrl2=l1/l2( here I denotes height of respective cone)
or,
the ratio of their surface area=4:3
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Two right circular cones have equal radii. If their slant heights are ...
Given:
- Two right circular cones
- Equal radii
To find:
- Ratio of their respective surface areas
Solution:
Let's assume that the radii of the cones are "r" and "r" respectively.
1. Slant Heights:
The slant height of a cone is the distance from the base to the apex along the slanted surface.
Given that the ratio of the slant heights is 4:3, we can write it as:
Slant height of the first cone = 4x
Slant height of the second cone = 3x
2. Surface Area:
The surface area of a cone is given by the formula:
Surface Area = πr(r + l), where r is the radius and l is the slant height.
Surface area of the first cone = πr(r + 4x)
Surface area of the second cone = πr(r + 3x)
3. Ratio of Surface Areas:
To find the ratio of the surface areas, we divide the surface area of the first cone by the surface area of the second cone:
Ratio = (Surface area of the first cone) / (Surface area of the second cone)
= (πr(r + 4x)) / (πr(r + 3x))
= (r + 4x) / (r + 3x)
4. Substitute the Values:
Since the radii of the cones are equal, we can substitute "r" for both the cones in the ratio:
Ratio = (r + 4x) / (r + 3x)
5. Simplify the Ratio:
To simplify the ratio, we can divide both numerator and denominator by "r":
Ratio = (1 + 4x/r) / (1 + 3x/r)
6. Substitute x = r:
Since the radii of the cones are equal, we can substitute "r" for "x" in the ratio:
Ratio = (1 + 4r/r) / (1 + 3r/r)
= (1 + 4) / (1 + 3)
= 5 / 4
Therefore, the ratio of their respective surface areas is 5:4, which can be written as 4:3. Thus, the correct answer is option D.
- Two right circular cones
- Equal radii
To find:
- Ratio of their respective surface areas
Solution:
Let's assume that the radii of the cones are "r" and "r" respectively.
1. Slant Heights:
The slant height of a cone is the distance from the base to the apex along the slanted surface.
Given that the ratio of the slant heights is 4:3, we can write it as:
Slant height of the first cone = 4x
Slant height of the second cone = 3x
2. Surface Area:
The surface area of a cone is given by the formula:
Surface Area = πr(r + l), where r is the radius and l is the slant height.
Surface area of the first cone = πr(r + 4x)
Surface area of the second cone = πr(r + 3x)
3. Ratio of Surface Areas:
To find the ratio of the surface areas, we divide the surface area of the first cone by the surface area of the second cone:
Ratio = (Surface area of the first cone) / (Surface area of the second cone)
= (πr(r + 4x)) / (πr(r + 3x))
= (r + 4x) / (r + 3x)
4. Substitute the Values:
Since the radii of the cones are equal, we can substitute "r" for both the cones in the ratio:
Ratio = (r + 4x) / (r + 3x)
5. Simplify the Ratio:
To simplify the ratio, we can divide both numerator and denominator by "r":
Ratio = (1 + 4x/r) / (1 + 3x/r)
6. Substitute x = r:
Since the radii of the cones are equal, we can substitute "r" for "x" in the ratio:
Ratio = (1 + 4r/r) / (1 + 3r/r)
= (1 + 4) / (1 + 3)
= 5 / 4
Therefore, the ratio of their respective surface areas is 5:4, which can be written as 4:3. Thus, the correct answer is option D.
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Two right circular cones have equal radii. If their slant heights are in the ratio 4 : 3, then their respective surface areas are in the ratioa)3 : 4b)16 : 9c)6 : 8d)4 : 3Correct answer is option 'D'. Can you explain this answer?
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