This problem requires us to determine the volume of a cylinder. We are provided with its base radius and a specific relationship between its curved surface area and total surface area. To solve this, we will use the standard geometric formulas for cylinders, first to find the cylinder's height, and then to calculate its volume.

Cylinder Problem Details

Let's carefully list the information given in the question:

• The base radius of the cylinder, denoted as \(r\), is 14 cm.
• The curved surface area (CSA) of the cylinder is three-fifth of its total surface area (TSA). This can be mathematically expressed as: \( \text{CSA} = \frac{3}{5} \times \text{TSA} \).
• Our objective is to find the volume (V) of the cylinder, which should be expressed in cubic centimeters (cu. cm).
• For calculations, we are instructed to use the value of \( \pi = \frac{22}{7} \).

Cylinder Formulas Used

To accurately solve this problem, we need to know the fundamental formulas related to a cylinder. Assuming \(r\) is the base radius and \(h\) is the height of the cylinder:

• Curved Surface Area (CSA): This represents the area of the lateral or curved part of the cylinder. The formula is: \( \text{CSA} = 2 \pi r h \)
• Total Surface Area (TSA): This is the sum of the curved surface area and the areas of both the top and bottom circular bases. The formula is: \( \text{TSA} = 2 \pi r h + 2 \pi r^2 \), which can also be written as \( \text{TSA} = 2 \pi r (r + h) \)
• Volume (V): This measures the space occupied by the cylinder. The formula is: \( \text{V} = \pi r^2 h \)

Calculating Cylinder Height

We are given a crucial relationship between the curved surface area and the total surface area:

\( \text{CSA} = \frac{3}{5} \times \text{TSA} \)

Let's substitute the formulas for CSA and TSA into this equation:

\( 2 \pi r h = \frac{3}{5} (2 \pi r (r + h)) \)

To simplify this equation and solve for the height \(h\), we can cancel out the common term \(2 \pi r\) from both sides of the equation (since \(r\) is a radius, it is not zero):

\( h = \frac{3}{5} (r + h) \)

Now, multiply both sides of the equation by 5 to eliminate the fraction:

\( 5h = 3(r + h) \)

Distribute the 3 on the right side of the equation:

\( 5h = 3r + 3h \)

To isolate the term with \(h\), subtract \(3h\) from both sides:

\( 5h - 3h = 3r \)

\( 2h = 3r \)

Finally, substitute the given base radius, \(r = 14\) cm, into this simplified equation:

\( 2h = 3 \times 14 \)

\( 2h = 42 \)

Divide by 2 to find the height \(h\):

\( h = \frac{42}{2} \)

\( h = 21 \text{ cm} \)

So, the height of the cylinder is 21 cm.

Volume Calculation of Cylinder

With both the base radius (\(r = 14\) cm) and the height (\(h = 21\) cm) determined, we can now calculate the volume of the cylinder using the formula: \( \text{V} = \pi r^2 h \).

Substitute the values along with \( \pi = \frac{22}{7} \):

\( \text{V} = \frac{22}{7} \times (14)^2 \times 21 \)

\( \text{V} = \frac{22}{7} \times 14 \times 14 \times 21 \)

We can simplify the expression by dividing 14 by 7:

\( \text{V} = 22 \times 2 \times 14 \times 21 \)

Now, perform the multiplication step-by-step:

\( \text{V} = 44 \times 14 \times 21 \)

\( \text{V} = 616 \times 21 \)

Performing the final multiplication of 616 by 21:

\( \text{V} = 12936 \text{ cu. cm} \)

Therefore, the volume of the cylinder is 12936 cubic centimeters.