The Geometry of Cylinders: Calculating Volume and Surface Area with Given Constraints

Understanding the geometric properties of cylinders and applying them to real-world scenarios can provide valuable insights into problem-solving techniques. In this article, we explore a specific problem involving a cylinder where the curved surface area is twice the area of its base, and the sum of the radius and height is 28 cm. We will derive the volume of the cylinder and provide a step-by-step solution using the given constraints.

Problem Statement

The problem at hand involves a cylinder where the curved surface area is twice the area of the base, and the sum of the radius and height is 28 cm. Our goal is to determine the volume of the cylinder.

Solution Steps

Step 1: Define Variables

Let's define the variables as follows: r radius of the cylinder h height of the cylinder

Step 2: Set Up Equations

From the problem, we have two pieces of information: The curved surface area (lateral surface area, LA) of the cylinder is twice the area of the base. The sum of the radius and height is 28 cm.

We can write these as equations as follows:

Curved surface area (LA) 2 × base area LA 2 × π × r × h Sum of radius and height 28 cm r h 28

Step 3: Solve for Height (h)

First, let's solve the second equation for h:

h 28 - r

Step 4: Substitute and Simplify

Next, substitute h 28 - r into the first equation:

2 × π × r × (28 - r) 2 × π × r^2

2πr(28 - r) 2πr^2

56πr - 2πr^2 2πr^2

56r - 2r^2 2r^2

56r - 4r^2 0

4r^2 - 56r 0

4r(r - 14) 0

This gives us:

r 0 (not applicable as radius cannot be zero) r 14 cm

Since the radius cannot be negative, the radius of the cylinder is 14 cm.

Step 5: Find the Height

Substitute r 14 cm into the second equation to find the height:

h 28 - 14 14 cm

Step 6: Calculate the Volume

Now, use the volume formula for a cylinder, V πr^2h:

V π × (14)^2 × 14

V π × 196 × 14

V 3234 cm^3 (using π ≈ 3.1416)

Conclusion

Therefore, the volume of the cylinder is 3234 cm^3. This solution demonstrates the application of fundamental geometric principles to find the solution to a complex problem.

Additional Insights and Practice

Practice similar problems to solidify your understanding and improve your problem-solving skills. Understanding the relationships between the different parts of a cylinder, such as the base area, lateral surface area, and height, is crucial in tackling these types of geometric problems. Always remember to check your work for inconsistencies and revalidate your steps to ensure accuracy.