9. If The Diameters Of Two Cylinders Are In The Ratio 3: 1 And The Ratio Of Their Heights Is 1: 3. Find The Ratio Of Their Volumes. (AHSEC 1995, 2003) [Ans. 3: 1]
Introduction
In mathematics, the volume of a cylinder is a fundamental concept that is used to calculate the amount of space inside a cylindrical object. The volume of a cylinder is given by the formula V = πr^2h, where r is the radius of the base and h is the height of the cylinder. In this article, we will discuss how to find the ratio of the volumes of two cylinders when their diameters and heights are given in a specific ratio.
Understanding the Problem
The problem states that the diameters of two cylinders are in the ratio 3:1 and the ratio of their heights is 1:3. We need to find the ratio of their volumes. To solve this problem, we will first understand the relationship between the diameter and the radius of a cylinder. The diameter of a cylinder is twice the radius, so if the diameters of two cylinders are in the ratio 3:1, then their radii will be in the ratio 3/2:1/2.
Calculating the Ratio of Radii
Let's assume that the radius of the first cylinder is 3x/2 and the radius of the second cylinder is x/2. Since the diameters are in the ratio 3:1, the radii will be in the ratio 3/2:1/2.
Calculating the Ratio of Heights
The problem states that the ratio of the heights of the two cylinders is 1:3. Let's assume that the height of the first cylinder is y and the height of the second cylinder is 3y.
Calculating the Ratio of Volumes
Now that we have the ratio of radii and heights, we can calculate the ratio of volumes. The volume of a cylinder is given by the formula V = πr^2h. Let's calculate the volume of the first cylinder and the second cylinder.
Volume of the First Cylinder
The volume of the first cylinder is V1 = π(3x/2)^2(y) = π(9x^2/4)(y) = (9πx^2y)/4.
Volume of the Second Cylinder
The volume of the second cylinder is V2 = π(x/2)^2(3y) = π(x^2/4)(3y) = (3πx^2y)/4.
Ratio of Volumes
Now that we have the volumes of the two cylinders, we can calculate the ratio of their volumes. The ratio of volumes is V1/V2 = ((9πx^2y)/4) / ((3πx^2y)/4) = 9/3 = 3:1.
Conclusion
In this article, we discussed how to find the ratio of the volumes of two cylinders when their diameters and heights are given in a specific ratio. We calculated the ratio of radii and heights, and then used the formula for the volume of a cylinder to calculate the ratio of volumes. The final answer is 3:1.
Key Takeaways
- The volume of a cylinder is given by the formula V = πr^2h.
- The ratio of radii is 3/2:1/2 when the diameters are in the ratio 3:1.
- The ratio of heights is 1:3 when the heights are in the ratio 1:3.
- The ratio of volumes is 3:1 when the diameters and heights are in the ratio 3:1 and 1:3 respectively.
Practice Problems
- If the diameters of two cylinders are in the ratio 2:3 and the ratio of their heights is 3:2, find the ratio of their volumes.
- If the diameters of two cylinders are in the ratio 4:5 and the ratio of their heights is 5:4, find the ratio of their volumes.
- If the diameters of two cylinders are in the ratio 6:7 and the ratio of their heights is 7:6, find the ratio of their volumes.
Solutions
- The ratio of radii is 2/3:1/3 when the diameters are in the ratio 2:3. The ratio of heights is 3:2 when the heights are in the ratio 3:2. The ratio of volumes is (2/3)^2(3/2) : (1/3)^2(2/2) = 4:1.
- The ratio of radii is 4/5:1/5 when the diameters are in the ratio 4:5. The ratio of heights is 5:4 when the heights are in the ratio 5:4. The ratio of volumes is (4/5)^2(5/4) : (1/5)^2(4/4) = 16:1.
- The ratio of radii is 6/7:1/7 when the diameters are in the ratio 6:7. The ratio of heights is 7:6 when the heights are in the ratio 7:6. The ratio of volumes is (6/7)^2(7/6) : (1/7)^2(6/6) = 36:1.
Q&A: Volume of Cylinders ==========================
Frequently Asked Questions
In this article, we will answer some frequently asked questions related to the volume of cylinders.
Q: What is the formula for the volume of a cylinder?
A: The formula for the volume of a cylinder is V = πr^2h, where r is the radius of the base and h is the height of the cylinder.
Q: How do I calculate the volume of a cylinder when the diameter and height are given?
A: To calculate the volume of a cylinder when the diameter and height are given, you need to first find the radius of the cylinder. The radius is half the diameter. Then, you can use the formula V = πr^2h to calculate the volume.
Q: What is the relationship between the diameter and the radius of a cylinder?
A: The diameter of a cylinder is twice the radius. So, if the diameter is given, you can find the radius by dividing the diameter by 2.
Q: How do I find the ratio of the volumes of two cylinders when their diameters and heights are given in a specific ratio?
A: To find the ratio of the volumes of two cylinders when their diameters and heights are given in a specific ratio, you need to first find the ratio of their radii and heights. Then, you can use the formula V = πr^2h to calculate the ratio of their volumes.
Q: What is the ratio of the volumes of two cylinders when their diameters are in the ratio 3:1 and the ratio of their heights is 1:3?
A: The ratio of the volumes of two cylinders when their diameters are in the ratio 3:1 and the ratio of their heights is 1:3 is 3:1.
Q: How do I calculate the volume of a cylinder when the radius and height are given?
A: To calculate the volume of a cylinder when the radius and height are given, you can use the formula V = πr^2h.
Q: What is the relationship between the volume of a cylinder and its radius and height?
A: The volume of a cylinder is directly proportional to the square of its radius and directly proportional to its height.
Q: How do I find the volume of a cylinder when the diameter and height are given in terms of a variable?
A: To find the volume of a cylinder when the diameter and height are given in terms of a variable, you need to first find the radius of the cylinder. Then, you can use the formula V = πr^2h to calculate the volume.
Q: What is the formula for the volume of a cylinder in terms of its diameter and height?
A: The formula for the volume of a cylinder in terms of its diameter and height is V = π(d/2)^2h, where d is the diameter of the cylinder.
Q: How do I calculate the volume of a cylinder when the diameter and height are given in a specific ratio?
A: To calculate the volume of a cylinder when the diameter and height are given in a specific ratio, you need to first find the ratio of their radii and heights. Then, you can use the formula V = πr^2h to calculate the volume.
Q: What is the relationship between the volume of a cylinder and its diameter and height?
A: The volume of a cylinder is directly proportional to the square of its diameter and directly proportional to its height.
Q: How do I find the ratio of the volumes of two cylinders when their diameters are in the ratio 2:3 and the ratio of their heights is 3:2?
A: The ratio of the volumes of two cylinders when their diameters are in the ratio 2:3 and the ratio of their heights is 3:2 is 4:1.
Q: What is the formula for the volume of a cylinder in terms of its radius and height?
A: The formula for the volume of a cylinder in terms of its radius and height is V = πr^2h.
Q: How do I calculate the volume of a cylinder when the radius and height are given in terms of a variable?
A: To calculate the volume of a cylinder when the radius and height are given in terms of a variable, you can use the formula V = πr^2h.
Q: What is the relationship between the volume of a cylinder and its radius and height in terms of a variable?
A: The volume of a cylinder is directly proportional to the square of its radius and directly proportional to its height in terms of a variable.