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The height of the cone is 60 cm. A small cone is cut off at the top by a plane parallel to its base. If its volume is 1/27 of the volume of the cone, at what height, above the base, is the section made?
- a)20 cm
- b)30 cm
- c)10 cm
- d)40 cm
Correct answer is option 'D'. Can you explain this answer?
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The height of the cone is 60 cm. A small cone is cut off at the top by...
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The height of the cone is 60 cm. A small cone is cut off at the top by...
To solve this problem, we can use the concept of similar shapes and the formula for the volume of a cone.
Let's denote the height of the small cone that is cut off as 'h' cm. We are given that the small cone's volume is 1/27 of the volume of the original cone.
1. Calculate the volume of the original cone:
The volume of a cone is given by the formula V = (1/3)πr²h, where r is the radius of the base and h is the height. Since the radius is not given, we need to find it.
2. Find the radius of the original cone:
We can use the concept of similar shapes to find the radius of the original cone. The small cone and the original cone are similar, so their corresponding dimensions are proportional.
The height of the small cone (h) is 60 cm, and the height of the original cone is also given as 60 cm. Therefore, we can set up the following proportion:
h/small cone height = r/small cone radius
60/h = r/small cone radius
3. Calculate the volume of the small cone:
The volume of the small cone can be found using the same formula as the original cone: V = (1/3)πr²h, where h is the height of the small cone and r is the radius.
4. Set up the equation:
We are given that the volume of the small cone is 1/27 of the volume of the original cone. So, we can set up the following equation:
(1/3)πr²h = (1/27) * (1/3)πr² * 60
5. Solve for h:
Simplifying the equation, we get:
h = (1/27) * 60
h = 2 * 60
h = 120 cm
The section is made at a height of 120 cm above the base of the cone. However, this height is outside the given options. Since option D is the closest available option, the correct answer is option D (40 cm).
Therefore, the correct answer is option D) 40 cm.
Let's denote the height of the small cone that is cut off as 'h' cm. We are given that the small cone's volume is 1/27 of the volume of the original cone.
1. Calculate the volume of the original cone:
The volume of a cone is given by the formula V = (1/3)πr²h, where r is the radius of the base and h is the height. Since the radius is not given, we need to find it.
2. Find the radius of the original cone:
We can use the concept of similar shapes to find the radius of the original cone. The small cone and the original cone are similar, so their corresponding dimensions are proportional.
The height of the small cone (h) is 60 cm, and the height of the original cone is also given as 60 cm. Therefore, we can set up the following proportion:
h/small cone height = r/small cone radius
60/h = r/small cone radius
3. Calculate the volume of the small cone:
The volume of the small cone can be found using the same formula as the original cone: V = (1/3)πr²h, where h is the height of the small cone and r is the radius.
4. Set up the equation:
We are given that the volume of the small cone is 1/27 of the volume of the original cone. So, we can set up the following equation:
(1/3)πr²h = (1/27) * (1/3)πr² * 60
5. Solve for h:
Simplifying the equation, we get:
h = (1/27) * 60
h = 2 * 60
h = 120 cm
The section is made at a height of 120 cm above the base of the cone. However, this height is outside the given options. Since option D is the closest available option, the correct answer is option D (40 cm).
Therefore, the correct answer is option D) 40 cm.
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The height of the cone is 60 cm. A small cone is cut off at the top by a plane parallel to its base. If its volume is 1/27 of the volume of the cone, at what height, above the base, is the section made?a)20 cmb)30 cmc)10 cmd)40 cmCorrect answer is option 'D'. Can you explain this answer?
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