Let's solve this geometry problem involving the curved surface area of a cylinder. We are given the curved surface area and the height of the cylinder, and we need to find the diameter of its base.

Understanding the Curved Surface Area of a Cylinder

The curved surface area (CSA) of a cylinder is the area of the lateral surface, excluding the top and bottom circular bases. It can be visualized as the area of a rectangle formed when the cylinder's curved surface is unrolled. The length of this rectangle is the circumference of the base ($2\pi r$), and the width is the height of the cylinder ($h$).

The formula for the curved surface area of a cylinder is:

$\text{CSA} = 2 \pi r h$

where:

• $r$ is the radius of the base
• $h$ is the height of the cylinder
• $\pi$ is a mathematical constant (approximately $22/7$ or $3.14159$)

Given Information and Goal

We are given the following values:

• Curved Surface Area (CSA) = 968 cm$^2$
• Height (h) = 11 cm

We need to find the diameter (d) of the base. Remember that the diameter is twice the radius ($d = 2r$).

Calculating the Radius of the Cylinder Base

We can use the CSA formula and plug in the given values to find the radius $r$. We'll use $\pi = 22/7$ for this calculation.

$\text{CSA} = 2 \pi r h$

Substitute the known values:

$968 = 2 \times \frac{22}{7} \times r \times 11$

Now, let's simplify the right side of the equation:

$968 = \frac{44}{7} \times r \times 11$

$968 = \frac{484}{7} \times r$

To find $r$, we need to isolate it. Multiply both sides by 7/484:

$r = 968 \times \frac{7}{484}$

We can simplify 968/484. Notice that $484 \times 2 = 968$.

$r = 2 \times 7$

$r = 14$

So, the radius of the base is 14 cm.

Finding the Diameter of the Cylinder Base

The diameter $d$ is twice the radius $r$.

$d = 2r$

Substitute the calculated radius:

$d = 2 \times 14$

$d = 28$

The diameter of the base of the cylinder is 28 cm.

Conclusion

Given the curved surface area of 968 cm$^2$ and a height of 11 cm, the diameter of the cylinder's base is 28 cm.

Revision Table: Cylinder Dimensions

Here's a summary of the values:

Additional Information: Cylinder Formulas

Beyond the curved surface area, there are other important formulas related to cylinders:

• Area of the base: The base is a circle, so its area is $\pi r^2$. Since there are two bases, their combined area is $2 \pi r^2$.
• Total Surface Area (TSA): This is the sum of the curved surface area and the areas of the two bases.

  $\text{TSA} = \text{CSA} + \text{Area of two bases}$

  $\text{TSA} = 2 \pi r h + 2 \pi r^2$

  $\text{TSA} = 2 \pi r (h + r)$

• Volume: The volume of a cylinder is the area of the base multiplied by the height.

  $\text{Volume} = (\text{Area of base}) \times h$

  $\text{Volume} = \pi r^2 h$

Understanding these formulas helps in solving various problems related to cylinders.