Understanding Cylinder Surface Areas

This question asks for the ratio of the total surface area to the curved surface area of a solid cylinder with given dimensions. A solid cylinder has three surfaces: the curved surface and two circular bases.

To solve this, we need to recall the formulas for the surface areas of a cylinder:

• Curved Surface Area (CSA)
• Total Surface Area (TSA)

Calculating Curved Surface Area (CSA)

The curved surface area of a cylinder is the area of the side that curves around. The formula is given by:

\(\text{CSA} = 2 \pi r h\)

Where:

• \(r\) is the radius of the base
• \(h\) is the height of the cylinder

Given radius \(r = 9\) cm and height \(h = 25\) cm, we can calculate the CSA:

\(\text{CSA} = 2 \times \pi \times 9 \times 25\)

\(\text{CSA} = 18 \times 25 \times \pi\)

\(\text{CSA} = 450 \pi\) sq cm

Calculating Total Surface Area (TSA)

The total surface area of a solid cylinder includes the curved surface area and the area of the two circular bases. The area of each circular base is \(\pi r^2\). So, the total surface area is:

\(\text{TSA} = \text{Curved Surface Area} + \text{Area of two bases}\)

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

Alternatively, we can factor out \(2 \pi r\):

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

Using the given radius \(r = 9\) cm and height \(h = 25\) cm:

\(\text{TSA} = 2 \times \pi \times 9 \times (25 + 9)\)

\(\text{TSA} = 18 \pi \times (34)\)

\(\text{TSA} = 612 \pi\) sq cm

Finding the Ratio of Total Surface Area to Curved Surface Area

The question asks for the ratio of TSA to CSA, which is \(\frac{\text{TSA}}{\text{CSA}}\).

\(\text{Ratio} = \frac{612 \pi}{450 \pi}\)

We can cancel out \(\pi\) from the numerator and the denominator:

\(\text{Ratio} = \frac{612}{450}\)

Now, we simplify the fraction by finding common factors. Both numbers are divisible by 2:

\(\frac{612 \div 2}{450 \div 2} = \frac{306}{225}\)

Both numbers are divisible by 3:

\(\frac{306 \div 3}{225 \div 3} = \frac{102}{75}\)

Both numbers are divisible by 3 again:

\(\frac{102 \div 3}{75 \div 3} = \frac{34}{25}\)

The simplified ratio is \(34 : 25\).

Verification using Formulas directly

We can also find the ratio directly using the formulas before substituting values:

\(\text{Ratio} = \frac{\text{TSA}}{\text{CSA}} = \frac{2 \pi r (r + h)}{2 \pi r h}\)

Cancel out \(2 \pi r\) from the numerator and the denominator:

\(\text{Ratio} = \frac{r + h}{h}\)

Substitute the given values \(r = 9\) cm and \(h = 25\) cm:

\(\text{Ratio} = \frac{9 + 25}{25} = \frac{34}{25}\)

This confirms the ratio is \(34 : 25\).

Conclusion

The ratio of the total surface area to the curved surface area of the solid cylinder with radius 9 cm and height 25 cm is \(34 : 25\).

Revision Table: Cylinder Formulas

Additional Information on Solid Geometry

Solid geometry deals with three-dimensional shapes, such as cylinders, cones, spheres, cubes, and cuboids. Understanding surface area and volume is crucial for these shapes.

• Surface Area: The total area of the outer surface of a 3D object. It is measured in square units (e.g., cm\(^2\), m\(^2\)).
• Volume: The amount of space occupied by a 3D object. It is measured in cubic units (e.g., cm\(^3\), m\(^3\)).
• Ratio: A comparison of two quantities of the same kind, usually expressed as a fraction or using a colon (e.g., a:b). Ratios are often simplified to their simplest form.

For a cylinder, the radius is the distance from the center of the circular base to its edge, and the height is the perpendicular distance between the two bases. In a solid cylinder, both bases are included when calculating the total surface area.