Understanding the Cylinder Problem: Finding Volume from CSA and Height

This problem asks us to find the volume of a cylinder given its curved surface area and height. We need to use the formulas for the curved surface area and the volume of a cylinder to solve this.

Here's what we are given:

• Curved Surface Area (CSA) of the cylinder = \(126 \pi \, \text{cm}^2\)
• Height (h) of the cylinder = \(14 \, \text{cm}\)

We need to find the Volume (V) of the cylinder.

Let's recall the relevant formulas for a cylinder with radius \(r\) and height \(h\):

• Curved Surface Area (CSA) = \(2 \pi r h\)
• Volume (V) = \(\pi r^2 h\)

Step-by-Step Calculation to Find Cylinder Volume

1. Calculate the Radius (r) using the Curved Surface Area

We are given the CSA and the height, so we can use the CSA formula to find the radius \(r\).

CSA = \(2 \pi r h\)

Substitute the given values into the formula:

\(126 \pi = 2 \pi r (14)\)

\(126 \pi = 28 \pi r\)

Now, solve for \(r\). We can divide both sides by \(28 \pi\):

\(r = \frac{126 \pi}{28 \pi}\)

Cancel out \(\pi\) from the numerator and denominator:

\(r = \frac{126}{28}\)

Simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 14:

\(r = \frac{126 \div 14}{28 \div 14}\)

\(r = \frac{9}{2} \, \text{cm}\)

So, the radius of the cylinder is \(4.5 \, \text{cm}\).

2. Calculate the Volume (V) using the Radius and Height

Now that we have the radius \(r = \frac{9}{2} \, \text{cm}\) and the height \(h = 14 \, \text{cm}\), we can calculate the volume using the volume formula:

V = \(\pi r^2 h\)

Substitute the values of \(r\) and \(h\) into the formula:

\(V = \pi \left(\frac{9}{2}\right)^2 (14)\)

\(V = \pi \left(\frac{81}{4}\right) (14)\)

Multiply the numbers:

\(V = \pi \left(\frac{81 \times 14}{4}\right)\)

We can simplify the fraction by dividing 14 by 2 and 4 by 2:

\(V = \pi \left(\frac{81 \times 7}{2}\right)\)

Now, multiply 81 by 7:

\(81 \times 7 = 567\)

So, the volume is:

\(V = \pi \left(\frac{567}{2}\right) \, \text{cm}^3\)

To express this as a mixed number, divide 567 by 2:

\(567 \div 2 = 283\) with a remainder of \(1\).

So, \(\frac{567}{2} = 283 \frac{1}{2}\).

Therefore, the volume of the cylinder is \(283 \frac{1}{2} \pi \, \text{cm}^3\).

Summary of Findings

Given the curved surface area and height of the cylinder, we first found the radius and then used it to calculate the volume.

• Radius (r) = \(\frac{9}{2} \, \text{cm}\)
• Height (h) = \(14 \, \text{cm}\)
• Volume (V) = \(283 \frac{1}{2} \pi \, \text{cm}^3\)

Revision Table: Cylinder Formulas

Additional Information: Properties of a Cylinder

A cylinder is a three-dimensional solid that holds two parallel bases, usually circular, connected by a curved surface. The line segment connecting the centers of the two bases is called the axis of the cylinder. If the axis is perpendicular to the bases, it is called a right cylinder. The height of the cylinder is the perpendicular distance between the bases. The radius of the cylinder is the radius of its circular base.