Calculating Cylinder Height from Surface Area and Volume

This problem asks us to find the height of a cylindrical object given its curved surface area and volume. We are provided with the curved surface area (CSA) as 88 cm² and the volume (V) as 132 cm³. We are also given the value of π as \(\frac{{22}}{7}\). To solve this, we will use the standard formulas for the curved surface area and volume of a cylinder.

Formulas for a Cylinder

Let 'r' be the radius of the base of the cylinder and 'h' be its height. The formulas are:

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

Setting Up Equations

Using the given information, we can write two equations:

Equation 1 (from CSA):

\(2\pi r h = 88\)

Equation 2 (from Volume):

\(\pi r^2 h = 132\)

Solving for Radius (r)

We can solve these two equations simultaneously to find the values of 'r' and 'h'. A common method is to divide the equation with the higher power of 'r' by the other equation. Let's divide Equation 2 by Equation 1:

\(\frac{\pi r^2 h}{2\pi r h} = \frac{132}{88}\)

Simplify the left side by cancelling common terms (\(\pi\), 'r', 'h'):

\(\frac{r}{2} = \frac{132}{88}\)

Now, simplify the fraction on the right side. Both 132 and 88 are divisible by 44 (since \(132 = 3 \times 44\) and \(88 = 2 \times 44\)):

\(\frac{r}{2} = \frac{3 \times 44}{2 \times 44}\)

\(\frac{r}{2} = \frac{3}{2}\)

Multiply both sides by 2 to find the radius 'r':

\(r = \frac{3}{2} \times 2\)

\(r = 3 \text{ cm}\)

Solving for Height (h)

Now that we have the radius \(r = 3\) cm, we can substitute this value into either Equation 1 or Equation 2 to find the height 'h'. Let's use Equation 1:

\(2\pi r h = 88\)

Substitute the values of \(\pi = \frac{22}{7}\) and \(r = 3\):

\(2 \times \frac{22}{7} \times 3 \times h = 88\)

\(\frac{44}{7} \times 3 \times h = 88\)

\(\frac{132}{7} \times h = 88\)

To isolate 'h', multiply both sides by \(\frac{7}{132}\):

\(h = 88 \times \frac{7}{132}\)

Simplify the fraction \(\frac{88}{132}\). Again, both are divisible by 44:

\(h = \frac{88}{132} \times 7\)

\(h = \frac{2 \times 44}{3 \times 44} \times 7\)

\(h = \frac{2}{3} \times 7\)

\(h = \frac{14}{3} \text{ cm}\)

Converting Height to Mixed Fraction

The calculated height is \(\frac{14}{3}\) cm. To match the format of the options, we can convert this improper fraction into a mixed fraction:

\(\frac{14}{3} = \frac{12 + 2}{3} = \frac{12}{3} + \frac{2}{3} = 4 + \frac{2}{3} = 4\frac{2}{3}\)

So, the height of the cylindrical object is \(4\frac{2}{3}\) cm.

Comparison with Options

Let's compare our calculated height with the given options:

• Option 1: 6 cm
• Option 2: \(4\frac{2}{3}\) cm
• Option 3: \(3\frac{2}{3}\) cm
• Option 4: 4 cm

Our calculated height \(4\frac{2}{3}\) cm matches Option 2.

Revision Table: Cylinder Height Calculation

Additional Information: Cylinder Properties

Understanding the properties of a cylinder is key to solving problems like this. Here are some related concepts:

• Cylinder: A three-dimensional solid that has two parallel circular bases connected by a curved surface.
• Radius (r): The distance from the center to any point on the circumference of the circular base.
• Height (h): The perpendicular distance between the two circular bases.
• Total Surface Area: The sum of the curved surface area and the areas of the two circular bases. Formula: \(2\pi r h + 2\pi r^2 = 2\pi r (h + r)\).
• Units: Ensure units are consistent. Area is measured in square units (cm²), and volume is measured in cubic units (cm³). Height and radius are measured in linear units (cm).
• Solving Simultaneous Equations: Problems involving multiple unknown variables often require setting up and solving a system of equations, as demonstrated here by dividing one equation by another.

By using the formulas correctly and performing algebraic manipulation, we successfully found the height of the cylindrical object.