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A cone and a cylinder have their heights in the ratio 4: 5 and their diameters are in the ratio 3: 2. The ratio of their volumes will be
- a)6: 7
- b)4: 3
- c)3: 5
- d)5: 3
Correct answer is option 'C'. Can you explain this answer?
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Verified Answer
A cone and a cylinder have their heights in the ratio 4: 5 and their d...
Height of cone =4x
Height of cylinder=5x
Diameter of cone=3y
Diameter of cylinder=2y
Volume of cone=
Volume of cylinder= πr2h=y2*5x
Height of cylinder=5x
Diameter of cone=3y
Diameter of cylinder=2y
Volume of cone=
Volume of cylinder= πr2h=y2*5xMost Upvoted Answer
A cone and a cylinder have their heights in the ratio 4: 5 and their d...
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A cone and a cylinder have their heights in the ratio 4: 5 and their d...
To find the ratio of the volumes of a cone and a cylinder, we need to compare their formulas for volume.
The formula for the volume of a cone is given by:
Vcone = (1/3)πr²h
The formula for the volume of a cylinder is given by:
Vcylinder = πr²h
Given that the heights of the cone and cylinder are in the ratio 4:5, we can assume that the height of the cone is 4x and the height of the cylinder is 5x.
Given that the diameters are in the ratio 3:2, we can assume that the diameter of the cone is 3y and the diameter of the cylinder is 2y. Therefore, the radius of the cone is (1/2)(3y) = (3/2)y and the radius of the cylinder is (1/2)(2y) = y.
Let's substitute these values into the formulas for volume.
Substituting the values for the cone:
Vcone = (1/3)π[(3/2)y]²(4x)
Vcone = (1/3)π(9/4)y²(4x)
Vcone = (3/4)πy²x
Substituting the values for the cylinder:
Vcylinder = πy²(5x)
Vcylinder = 5πy²x
Now we can compare the ratios by dividing the volume of the cone by the volume of the cylinder:
(Vcone) / (Vcylinder) = ((3/4)πy²x) / (5πy²x)
= (3/4) / 5
= 3/20
Therefore, the ratio of the volumes of the cone and cylinder is 3:20. However, this is not one of the given options.
To simplify the ratio, we can divide both terms by 3:
(3/3) / (20/3) = 1 / (20/3)
So, the simplified ratio is 1: (20/3).
Multiplying the numerator and denominator of the fraction by 3:
1 * 3 / (20/3) * 3 = 3 / (20/1)
Thus, the simplified ratio is 3:20/1, which is equivalent to 3:20.
The formula for the volume of a cone is given by:
Vcone = (1/3)πr²h
The formula for the volume of a cylinder is given by:
Vcylinder = πr²h
Given that the heights of the cone and cylinder are in the ratio 4:5, we can assume that the height of the cone is 4x and the height of the cylinder is 5x.
Given that the diameters are in the ratio 3:2, we can assume that the diameter of the cone is 3y and the diameter of the cylinder is 2y. Therefore, the radius of the cone is (1/2)(3y) = (3/2)y and the radius of the cylinder is (1/2)(2y) = y.
Let's substitute these values into the formulas for volume.
Substituting the values for the cone:
Vcone = (1/3)π[(3/2)y]²(4x)
Vcone = (1/3)π(9/4)y²(4x)
Vcone = (3/4)πy²x
Substituting the values for the cylinder:
Vcylinder = πy²(5x)
Vcylinder = 5πy²x
Now we can compare the ratios by dividing the volume of the cone by the volume of the cylinder:
(Vcone) / (Vcylinder) = ((3/4)πy²x) / (5πy²x)
= (3/4) / 5
= 3/20
Therefore, the ratio of the volumes of the cone and cylinder is 3:20. However, this is not one of the given options.
To simplify the ratio, we can divide both terms by 3:
(3/3) / (20/3) = 1 / (20/3)
So, the simplified ratio is 1: (20/3).
Multiplying the numerator and denominator of the fraction by 3:
1 * 3 / (20/3) * 3 = 3 / (20/1)
Thus, the simplified ratio is 3:20/1, which is equivalent to 3:20.
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A cone and a cylinder have their heights in the ratio 4: 5 and their diameters are in the ratio 3: 2. The ratio of their volumes will bea)6: 7b)4: 3c)3: 5d)5: 3Correct answer is option 'C'. Can you explain this answer?
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