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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)
    5:3
  • b)
    6:7
  • c)
    3:5
  • d)
    4:3
Correct answer is option 'C'. Can you explain this answer?
Most Upvoted Answer
A cone and a cylinder have their heights in the ratio 4: 5 and their d...
Understanding the Problem
To find the ratio of the volumes of a cone and a cylinder, we must first understand the formulas for their volumes and the provided ratios.
Volume Formulas
- Volume of a cone: V_cone = (1/3) * π * r² * h
- Volume of a cylinder: V_cylinder = π * r² * h
Given Ratios
- Height ratio (Cone: Cylinder) = 4:5
- Diameter ratio (Cone: Cylinder) = 3:2
Since diameter is twice the radius, the radius ratio will also be:
- Radius ratio (Cone: Cylinder) = 3/2 : 2/2 = 3:4
Assigning Variables
Let's assign values based on the ratios:
- Height of cone (h_c) = 4k
- Height of cylinder (h_cyl) = 5k
- Radius of cone (r_c) = 3m
- Radius of cylinder (r_cyl) = 4m
Calculating Volumes
Now, we can substitute these values into the volume formulas.
- Volume of cone:
V_cone = (1/3) * π * (3m)² * (4k)
= (1/3) * π * 9m² * 4k
= 12πmk²
- Volume of cylinder:
V_cylinder = π * (4m)² * (5k)
= π * 16m² * 5k
= 80πmk²
Finding the Volume Ratio
Now, we can find the ratio of the volumes:
- Ratio = V_cone : V_cylinder
= 12πmk² : 80πmk²
= 12 : 80
= 3 : 20
This simplifies to 3:5.
Conclusion
The ratio of the volumes of the cone to the cylinder is indeed 3:5, confirming that option 'C' is correct.
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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)5:3b)6:7c)3:5d)4:3Correct answer is option 'C'. Can you explain this answer?
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