Calculating the Cost of Polishing a Cylinder's Curved Surface

The problem asks us to find the total cost of polishing the curved surface of a wooden cylinder. We are given the dimensions of the cylinder and the rate at which the polishing is done per square meter.

Understanding the Given Information

• Shape to be polished: Curved surface of a cylinder.
• Diameter of the cylinder: 70 cm.
• Height of the cylinder: 6 m.
• Polishing rate: Rs. 50 per m².
• Value of π to use: \(\frac{22}{7}\).

Step 1: Ensure Units are Consistent

The polishing rate is given in Rupees per square meter (m²), but the diameter is given in centimeters (cm). We need to convert the diameter into meters so that all measurements are in the same unit (meters).

Diameter = 70 cm

Since 1 m = 100 cm, we convert cm to meters by dividing by 100.

Diameter in meters = \(\frac{70}{100}\) m = 0.70 m

The height is already in meters.

Height (h) = 6 m

Step 2: Find the Radius of the Cylinder

The radius of a cylinder is half of its diameter.

Radius (r) = \(\frac{\text{Diameter}}{2}\)

Radius (r) = \(\frac{0.70 \text{ m}}{2}\) = 0.35 m

Step 3: Calculate the Curved Surface Area (CSA) of the Cylinder

The area to be polished is the curved surface area of the cylinder. The formula for the curved surface area of a cylinder is:

CSA = \(2 \pi r h\)

Substitute the values of π, r, and h into the formula:

CSA = \(2 \times \frac{22}{7} \times 0.35 \text{ m} \times 6 \text{ m}\)

Let's simplify the calculation:

CSA = \(2 \times \frac{22}{7} \times \frac{35}{100} \times 6\) m²

We can simplify \(\frac{35}{7}\) to 5:

CSA = \(2 \times 22 \times \frac{5}{100} \times 6\) m²

Now simplify \(\frac{5}{100}\) to \(\frac{1}{20}\):

CSA = \(2 \times 22 \times \frac{1}{20} \times 6\) m²

Simplify \(2 \times \frac{1}{20}\) to \(\frac{1}{10}\):

CSA = \(22 \times \frac{1}{10} \times 6\) m²

CSA = \(22 \times 0.1 \times 6\) m²

CSA = \(2.2 \times 6\) m²

CSA = 13.2 m²

So, the curved surface area of the wooden cylinder is 13.2 square meters.

Step 4: Calculate the Total Cost of Polishing

The cost of polishing is Rs. 50 per square meter. To find the total cost, we multiply the curved surface area by the polishing rate.

Total Cost = Curved Surface Area \(\times\) Rate per m²

Total Cost = \(13.2 \text{ m}^2 \times 50 \text{ Rs./m}^2\)

Total Cost = \(13.2 \times 50\) Rs.

Total Cost = \(132 \times 5\) Rs. (multiplying 13.2 by 10 and dividing 50 by 10)

Total Cost = 660 Rs.

Conclusion

The total cost of polishing the curved surface of the wooden cylinder is Rs. 660.

Revision Table: Cylinder Surface Area & Cost Calculation

Additional Information: Properties of a Cylinder

A cylinder is a 3D shape with two parallel circular bases connected by a curved surface. Key properties include:

• It has three faces: the top circular base, the bottom circular base, and the curved lateral surface.
• It has two edges, which are the circumferences of the circular bases.
• It has no vertices (corners).
• The height is the perpendicular distance between the two bases.
• The radius is the radius of the circular bases.

Calculating surface areas and volumes of cylinders is a common topic in mensuration, which deals with the measurement of geometric shapes.