A company produces cylindrical cans with a height of 12 cm and a radius of 5 cm. If the company wants to paint the entire outer surface of each can including the top and bottom, how much paint is needed if the paint covers 100 square centimeters per liter and the paint costs $15 per liter?

["Optimizing Paint Usage: Calculating Materials for a Company’s Cylindrical Cans", "In today’s packaging industry, precision and cost-efficiency are key—especially when coating cylindrical cans that require full surface coverage. A company producing cylindrical cans with a height of 12 cm and a radius of 5 cm faces the practical challenge of determining exactly how much paint is needed to cover each can completely. This article breaks down the surface area calculations, paint requirements, and cost analysis to help manufacturers optimize material use and production budgets.", "---", "### Understanding the Cylindrical Can’s Surface Area", "The total outer surface area of a cylinder includes both the lateral (side) surface and the two circular top and bottom ends. The formula for the total surface area ( A ) of a cylinder is:", "[\nA = 2\pi r h + 2\pi r^2\n]", "Where:\n- ( r ) = radius (5 cm)\n- ( h ) = height (12 cm)\n- ( \pi \approx 3.1416 )", "---", "### Step-by-Step Surface Area Calculation", "1. Calculate the lateral surface area:", "[\n2\pi r h = 2 \ imes \pi \ imes 5 \ imes 12 = 120\pi \approx 376.99 , \ ext{cm}^2\n]", "2. Calculate the area of the two circular ends:", "[\n2\pi r^2 = 2 \ imes \pi \ imes 5^2 = 2 \ imes \pi \ imes 25 = 50\pi \approx 157.08 , \ ext{cm}^2\n]", "3. Total surface area:", "[\nA = 376.99 + 157.08 = 534.07 , \ ext{cm}^2\n]", "---", "### Determining Paint Volume Required", "Since the paint covers 100 square centimeters per liter, the total paint needed is:", "[\n\ ext{Paint required (in liters)} = \frac{534.07}{100} = 5.34 , \ ext{liters}\n]", "Because paint must be purchased in full liters, the company will need to buy 6 liters to ensure full coverage.", "---", "### Cost Analysis: How Much Will the Paint Cost?", "At $15 per liter, the total paint cost is:", "[\n\ ext{Total cost} = 6 \ imes 15 = $90\n]", "This cost highlights the importance of accurate surface measurement—not only for minimizing waste but also for controlling expenses in large-scale production.", "---", "### Conclusion", "For a company manufacturing cylindrical cans with 12 cm height and 5 cm radius, coating the entire outer surface requires approximately 534.07 cm² of paint. With a coverage rate of 100 cm² per liter, this equates to 6 liters of paint, costing $90. These calculations empower manufacturers to improve procurement accuracy, reduce material waste, and maintain tight production budgets—critical elements in today’s competitive packaging market.", "---", "Keywords: cylindrical cans, paint coverage, surface area calculation, paint volume, manufacturing costs, can coating, paint usage, cylindrical can production, packing materials, cost efficiency."]