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The height of a right circular cone is trisected by two planes parallel to its base at equal distances. The volumes of the three solids, so obtained, starting from the top, are in the ratio:
- a)1 : 8 : 28
- b)1 : 7 : 19
- c)More than one of the above
- d)None of the above
Correct answer is option 'B'. Can you explain this answer?
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Most Upvoted Answer
The height of a right circular cone is trisected by two planes paralle...
Explanation:
Given:
The height of a right circular cone is trisected by two planes parallel to its base at equal distances.
Volume Ratios:
Let the height of the cone be h.
The volumes of the three solids formed are in the ratio of 1:7:19.
Volume of Top Cone:
The top cone has a height of h/3 and its volume is given by V1 = 1/3 * π * r^2 * h/3, where r is the radius of the cone.
Volume of Middle Cone:
The middle cone has a height of h/3 and its volume is given by V2 = 1/3 * π * r^2 * h/3, where r is the radius of the cone.
Volume of Bottom Cone:
The bottom cone has a height of h/3 and its volume is given by V3 = 1/3 * π * r^2 * h/3, where r is the radius of the cone.
Volume Ratio Calculation:
V1 : V2 : V3 = 1/3 * π * r^2 * h/3 : 1/3 * π * r^2 * h/3 : 1/3 * π * r^2 * h/3
= 1 : 1 : 1
Therefore, the volumes of the three solids are in the ratio 1:1:1.
Conclusion:
The given volume ratios are incorrect. The correct volume ratios of the three solids formed by trisecting the height of the cone are 1:1:1.
Given:
The height of a right circular cone is trisected by two planes parallel to its base at equal distances.
Volume Ratios:
Let the height of the cone be h.
The volumes of the three solids formed are in the ratio of 1:7:19.
Volume of Top Cone:
The top cone has a height of h/3 and its volume is given by V1 = 1/3 * π * r^2 * h/3, where r is the radius of the cone.
Volume of Middle Cone:
The middle cone has a height of h/3 and its volume is given by V2 = 1/3 * π * r^2 * h/3, where r is the radius of the cone.
Volume of Bottom Cone:
The bottom cone has a height of h/3 and its volume is given by V3 = 1/3 * π * r^2 * h/3, where r is the radius of the cone.
Volume Ratio Calculation:
V1 : V2 : V3 = 1/3 * π * r^2 * h/3 : 1/3 * π * r^2 * h/3 : 1/3 * π * r^2 * h/3
= 1 : 1 : 1
Therefore, the volumes of the three solids are in the ratio 1:1:1.
Conclusion:
The given volume ratios are incorrect. The correct volume ratios of the three solids formed by trisecting the height of the cone are 1:1:1.
Community Answer
The height of a right circular cone is trisected by two planes paralle...
The height of a right circular cone is trisected by two planes parallel to its base at equal distances.
Formula used:
Cone:
The volume of frustum
Calculation:
According to the question, the required figure is:
Now,
The volume of the cone AB''D'',
The volume of frustum cone B'D'BD,
The required ratio
∴ 1 : 7 : 19 is the requried ratio.
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The height of a right circular cone is trisected by two planes parallel to its base at equal distances. The volumes of the three solids, so obtained, starting from the top, are in the ratio:a)1 : 8 : 28b)1 : 7 : 19c)More than one of the aboved)None of the aboveCorrect answer is option 'B'. Can you explain this answer?
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